1

-SELFE Users Trainng Course-(1)

SELFE physical formulation

Joseph Zhang, CMOP, OHSU

2

Continuity equation

Momentum equations

Vertical b.c.:

Equation of state

Transport of salt and temperature

Turbulence closure: Umlauf and Burchard 2003

Governing equations: Reynolds-averged Navier-Stokes

0, ( ( , ))w

u vz

u u

),,( TSp

hdz

t0u

( ), ( , )h

Dc cQ c c S T

Dt z z

0 0

1ˆ + ; ( )A z

D gg f g p d

Dt z z

u uf f k u u

at =w zz

u

τ

| | , at D b bC z hz

u

u u

MSL

z

xEast

shear stratification

22 NKMKz

k

zDt

Dkhvmvk

wallhvmv FcNKcMKc

kzzDt

D2

23

21

,0 nmpkc

3Continuity equation (1)

Law of mass conservation + incompressibility of water Vertical integration

x

yz

z

x yu

w

v

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

y z x x u x x x u xV t

x z y y v y y y v y x y z z w z z z w z

( )V x y z

3

( ) ( ) ( )

( )

u v w

t x y z

u

3 3( ) 0D

t Dt

u u

( , 0)t V Eulerian form:

Lagrangian form:

Boussinesq:

0 3 0 u (Volume conservation)

0hh

dz w

u

4Continuity equation (2)

0hh

dz w

u

( , , )

x y t

z x y t

dz dx dyw u v

dt dt x dt y t

Vertical boundary conditions: kinematic z

surface

bottom( , )

x y

z h x y

dzw uh vh

dt

Integrated continuity equation

0

( )

t x y x yh

h h

dz u v uh vh

dz dz h

u

u u u

0h

dzt

u

5Momentum equation (1)

3 3

1 + ( )

1+ ( )

DP f

Dt

Dw Pg w

Dt z

uu k u

mF aNewton’s 2nd law (Lagrangian)

>

f

Hydrostatic assumption1

0 Az

Pg P g d P

z

Separation of scales1

( ) + ( )Az

Dd P f

Dt z z

u uu k u

Boussinesq assumption

z zd d

0 0

1ˆ + ( )A z

D gg f g p d

Dt z z

u uk u u

6

Coriolis

Coriolis: “virtual force” Newton’s 2nd law is only valid in an inertial frame For large scale GFD flows, the earth is not an inertial frame Small as it is, it is reponsible for a number of unique

phenomena in GFD Accelerations in a rotating frame ( const.):

I

Jij

u

xr

~~~~ruU

~~

~

dtdt :or r

dRd

~~~~~~~~~

2dt

ruaUd

A

Coriolis

Centrifugal acceleration

7

Coriolis Plane coordinates: neglect curvature of the earth

x: east; y: north; z: vertical upward Projection of earth rotation in the local frame:

Components of Coriolis acceleration:

Inertial oscillation: no external forces, advection and vertical vel.

e.g., sudden shut-down of wind in ocean.

x

y z

~~~sincos kje

sin2 ,cos2

:

:

:

*

*

*

ff

ufz

fuy

fvwfx

)cos(

)sin(

0

0

ftVv

ftVu

fudt

dv

fvdt

du

0~

u

)

8

Momentum equation: vertical boundary condition (b.c.)

at =w zz

u

τ | | , at D b bC z hz

u

u uSurface Bottom Logarithmic law:

Reynolds stress:

Turbulence closure:

Reynolds stress (const.)

Drag coefficient:

00

0

0

ln ( ) /, ( )

ln( / )

: bottom roughness

b bb

z h zz h z h

z

z

u u

0( ) ln( / ) bbz z h z

u

u

1/ 2

2

2 / 3 21

0

2 ,

,

1| |

2( )

m

m

D b

s K l

s g

K B C

l z h

u

1/ 200

0

| | , ( )ln( / ) D b b b

b

C z h z hz z

u

u u

2

0 0

1ln b

DCz

b

ub

z=h

9Momentum equation (3)

Implications• Use of z0 seems most natural option, but over-estimates CD in shallow area• Strictly speaking CD should vary with bottom layer thickness (i.e. vertical grid)• Different from 2D, 3D results of elevation depend on vertical grid (with constant CD)

meters

10Transport equation (1)

( ), ( , )h

Dc cQ c c S T

Dt z z

Mass conservation: advection-diffusion equation

),,( TSp

Fick’s law for diffusive fluxes

Advection

Diffusion

>

B.c.

, ˆ

0,

cc z

zc

z hn

xq y z x

xq

q x y zx

c

t

x y z

11Transport equation (2)

0

( ),

S S E Pz

z

Precipitation and evaporation model

E: evaporation rate (kg/m2/s) (either measured or calculated)P: Precipitation rate (kg/m2/s) (usually measured)

Heat exchange model

0

, p

T Hz

z C

( )H IR IR S E

IR’s are down/upwelling infrared (LW) radiation at surfaceS is the turbulent flux of sensible heat (upwelling)E is the turbulent flux of latent heat (upwelling)SW is net downward solar radiation at surface

The solar radiation is penetrative (body force) with attenuation (which depends upon turbidity) acts as a heat source within the water.

0

1

p

SWQ

C z

12Air-water exchange

1. free surface height: variations in atmospheric pressure over the domain have a direct impact upon free surface height; set up due to wind stresses

2. momentum: near-surface winds apply wind-stress on surface (influences advection, location of density fronts)

3. heat: various components of heat fuxes (dependent upon many variables) determine surface heat budget• shortwave radiation (solar) - penetrative• longwave radiation (infrared)• sensible heat flux (direct transfer of heat)• latent heat flux (heating/cooling associated with

condensation/evaporation)4. water: evaporation, condensation, precipitation act as

sources/sinks of fresh water

13

Downwelling SW at the surface is forecast in NWP models - a function of time of year, time of day, weather conditions, latitude, etcUpwelling SW is a simple function of downwelling SW

Solar radiation

SW SW

is the albedo - typically depends on solar zenith angle and sea state

Attenuation of SW radiation in the water column is a function of turbidity and depth d (Jerlov, 1968, 1976; Paulson and Clayson, 1977):

1 2/ /( ) (1 ) Re (1 R)eD d D dSW z SW R, d1 and d2 depend on water type; D is the distance from F.S.

Type R d1 (m) d2 (m)

Jerlov I 0.58 0.35 23

Jerlov IA 0.62 0.60 20

Jerlov IB 0.67 1.00 17

Jerlov II 0.77 1.50 14

Jerlov III 0.78 1.40 7.9

Paulson and Simpson

0.62 1.50 20

Estuary 0.80 0.90 2.1

SW

Depth

(m

)

I

II

III

14Infrared radiation

• Downwelling IR at the surface is forecast in NWP models - a function of air temperature, cloud cover, humidity, etc• Upwelling IR is can be approximated as either a broadband measurement solely within the IR wavelengths (i.e., 4-50 μm), or more commonly the blackbody radiative flux

is the emissivity, ~1 is the Stefan-Boltzmann constantTsfc is the surface temperature

4sfcIR T

15Turbulent Fluxes of Sensible and Latent Heat

In general, turbulent fluxes are a function of:• Tsfc,, Tair

• near-surface wind speed• surface atmospheric pressure• near-surface humidity

Scales of motion responsible for these heat fluxes are much smaller than canbe resolved by any operational model - they must be parameterized (i.e., bulk aerodynamic formulation)

where:u* is the friction scaling velocityT* is the temperature scaling parameterq* is the specific humidity scaling parametera is the surface air densityCpa is the specific heat of airLe is the latent heat of vaporization

The scaling parameters are defined using Monin-Obukhov similarity theory,and must be solved for iteratively (i.e., Zeng et al., 1998).

* *

* *

' '

' '

a pa a pa

a e a e

S C w T C u T

E L w q L u q

16Wind Shear Stress: Turbulent Flux of Momentum

Calculation of shear stress follows naturally from calculation of turbulent heat fluxes

Total shear stress of atmosphere upon surface:2*

0

aw u

Alternatively, Pond and Picard’s formulation can be used as a simpler option

0

| |aw ds w wC

τ u u

'

'

0.61 0.063

1000

max(6,min(50, ))

wds

w w

uC

u u

17Inputs to heat/momentum exchange model• Forecasts of uw vw @ 10m, PA @ MSL,Tair and qair @2m

• used by the bulk aerodynamic model to calculate the scaling parameters

• Forecasts of downward IR and SW

data sourcesupplying

agencytime period spatial resolution

temporal resolution

area of coverage data type

MRF NCEP04/01/2001-02/29/2004

1° x 1°12-hour

snapshots129W-120W, 35N-51N forecast

GFS NCEP 07/03/2003-present 1° x 1° 3-hour snapshots 180W-70W, 90S-90N forecast

OSU-ARPS OSU05/04/2001-02/25/2004

12 km 1-hour snapshots128W-119W, 41N-47N

(approx)forecast

ETA/NAM NCEP 07/03/2003-present 12 km 3-hour snapshotsEta

Grid 218, west of 100Wforecast

NCAR/NCEP Reanalysis (NARR)

NCAR01/01/1979-12/31/2006

12km 6-hour snapshots North America reanalysis

Atmospheric properties

data sourcesupplying

agencytime period spatial resolution

temporal resolution

area of coverage data type

AVN (lo-res) NCEP04/01/2001-10/28/2002

0.7° x 0.7° (approx) 3-hour averages 129W-120W, 35N-51N forecase

AVN (hi-res) NCEP10/29/2002-02/29/2004

0.5° x 0.5° (approx) 3-hour averages 129W-120W, 35N-51N forecast

GFS NCEP 07/03/2003-present 0.5° x 0.5° (approx) 3-hour averages 180W-70W, 90S-90N forecast

ETA/NAM NCEP 07/03/2003-present 12 km 3-hour snapshotsEta Grid 218, west of 100W

forecast

NCAR/NCEP Reanalysis (NARR)

NCAR01/01/1979-12/31/2006

12km 6-hour averages North Americareanalysi

s

Heat fluxes

18Turbulence closure: Pacanowski and Philander (1982)

02 2

buoyancysheari

gz

Ru vz z

• Kelvin-Helmholtz instability: occurs at interface of two fluids or fluids of different density (stratification)• When stratification is present, the onset of instability is govern by the Richardson number

• Theory predicts that the fluid is unstable when Ri<0.25

maxmin2

min

(1 5 )

1 5

i

i

R

R

19

2 2k

Dk kM N

Dt z z

Turbulence closure: Mellor-Yamada-Galperin (1988)

2 21( ) ( )2 2k

lED kl kl lM N

Dt z z

• Theorize that the turbulence mixing is related to the growth and dissipation of turbulence kinetic energy (k) and Monin-Obukhov mixing length (l)• k and l are derived from the 2nd moment turbulence theory

3/ 2

1

(2 )kB l

Dissipation rate

Shear & buoyancy freq2 2

2 u vM

z z

2

0

gN

z

Viscosity and diffusivities1/ 2(2 )ms k l 1/ 2(2 )hs k l

1/ 20.2(2 )k k l

Stability functions 2 3

4 5(1 )(1 )h

mh h

g g Gs

g G g G

6

41hh

gs

g G

2

0

0.28 0.02332h

l gG

k z

Galperin clipping

Wall proximity function2 2

2 30 0

1b s

l lE E

d d

20Turbulence closure: Mellor-Yamada-Galperin (1988)

• Vertical b.c.• TKE related to stresses• Mixing length related to the distance from the “walls” (note the singularity)

2/31

0

2

/

Bk

z

l d

u

• Initial condition

6 2

8

5 10 m / s

=10

k

21

2 2k

Dk kM N

Dt z z

Turbulence closure: Umlauf and Burchard (2003)

• Use a generic length-scale variable to represent various closure schemes

• Mellor-Yamada-Galperin (with modification to wall-proximity function k-kl)• k- (Rodi)• k- (Wilcox)• KPP (Large et al.; Durski et al)

2 21 3 2 wall

Dc M c N c F

Dt z z k

0 p m nc k

30 3/ 2 1c k

1/ 2c k ' 1/ 2c k kk

Stability: Kantha and Clayson (smoothes Galperin’s stability function as it approaches max)

Diffusivity

'2 , 2

0.49391 30.19

m h

hh

c s c s

sG

0.392 17.07

1 6.127h h

mh

s Gs

G

2

_ _ _

_ 0 _

_

( )0.28 0.023

2

0.02

h u h u h ch

h u h h c

h c

G G GG

G G G

G

Wall function2 2

2 40 0

1wallb s

l lF E E

d d

22Turbulence closure: Umlauf and Burchard (2003)

2

3 23

1, 0

, 0

Nc

c N

Constants

Wall proximity function

3c depends on choice of scheme and

stability function

23Geophysical fluid dynamics in ocean

Coriolis plays an important role in GFD

Although Ek<<1, the friction term is important in boundary layer, and is the highest-order term in the eq.

Reynolds number:

Eddy viscosity >> molecular viscosity

1 1 1

1 1

1

20

20

22

0

kooT ERR

HLUρ

P

UH

WL

L

U

L

U

T

H

U

Lρ

PU

H

UW

L

U

L

U

T

U

z

u

zx

pfv

z

uw

y

uv

x

uu

t

u

12

H

L

E

RULR

k

oe

Ekman and Rossby numbers

24Geostrophy

Rapidly rotating (Coriolis dominant), homogeneous inviscid fluids:

N-S eq:

Taylor-Proudman theorem (vertical rigidity):

Flow is along isobars, i.e., streamline = isobar No work done by pressure; no energy required to sustain the flow

(inertial) Direction along the isobars

0 ,1 , , kooT ERR

0

0

0

1, ( )

1, ( )

10 , ( )

0, ( )

pfv a

x

pfu b

y

pc

z

d

u

0

z

v

z

u

highhigh

low

<

>

<

>

<low

>>

~u

p

25

Surface Ekman layer

EL exists whenever viscosity/shear is important Steady, homogeneous, uniform interior:

Solution:

zwind

EL d

interior),( vu

, ,

0at , ,

,)(

,)(

00

2

2

2

2

zvvuu

zdz

dv

dz

dudz

vduuf

dz

udvvf

yx

fd

d

z

d

ze

fdvv

d

z

d

ze

fduu

yxdz

yxdz

2

4cos

4sin

2

4sin

4cos

2

/

0

/

0

26

Ekman transport Properties:

Velocities can be very large for nearly inviscid fluid Wind-generated vel. makes a 45o angle to the wind direction Ekman spiral Net horizontal transport:

Ekman transport is perpendicular to the direction of wind stress (to the right in the Northern Hemisphere)

Bottom Ekman layer

) 45o

wind Surface current

Ekman transport

0),(),(

,1

)(

,1

)(

0

0

0

0

yx

x

y

VU

fdzvvV

fdzuuU

wind

Surface currentz

Ekman spira

l

27Barotropic waves Linear wave theory

2D monochromatic wave:

Fixed (x,y), varying t: T=2 T=2/(period)

||

2 :length wave

.const :line phase

phase reference :

frequencyangular :

tornumber vec wave:)(

amplitude :

)cos(

~

~

)(

tmylx

l,m

A

AeRtmylxA tmylxie

x

y

crest

trough

L1

L2

=A=0 =A=0=A

t fixed

T

(x,y) fixed

28

Phase speed Vary (x,y,t), but follow a phase line L: lx+my-t=C. The speed of a phase

In many dynamic systems, and are related in the dispersion relation:

e.g., surface gravity waves: Dispersion: a group of waves with different wavelength travel

with different speed Non-dispersive waves: tides

~~

ofdirection in the ||

c

)(~

hg tanh

29Group velocity

Wave energy ~ A2

Two 1D waves of equal amplitude and almost equal wave numbers:

The speed at which the envelop/energy travels is the group velocity

)cos(22

cos2

|,||| , ,2

|,||| , ,2

),2,1( )(

),cos()cos(

2121

2121

2211

txtxA

i

txAtxA

iii

average wave

Modulation/envelop of average wave

freq. beating :

d

dcg

Cg

C

’

30Linearization

Linearization of advective terms: Ro << 1 Shallow water model:

For flat bottom, h=+H (H: mean depth). Linearizing for small-amplitude waves ~A<<H leads to:

(a-c) govern the dynamics of linear barotropic waves on a flat bottom Neglecting Coriolis, the 1D wave equation is recovered

0)()(

(b)

(a)

y

hv

x

hu

t

h

ygfu

t

vx

gfvt

u

z

O

h(x,y,t)

b(x,y)

(wave continuity equation)

(c) 0

y

v

x

uH

t

tt xxgH phase speed pC gH

31Kelvin waves Semi-infinite domain of flat bottom, free slip at coast (a-c) become (3 eqs. for 2 unknowns; u=0):

Solution:

where F is an arbitrary function Properties:

Kelvin wave is trapped near the coast; Rossby radius of deformation R is a measurement of trapping;

The wave travels alongshore w/o deformation, at the speed of surface gravity waves c, with the coast on its right in the N-H;

The direction of the alongshore current (v) is arbitrary: Upwelling wave >0 v<0, same direction as phase speed; Downwelling wave <0 v>0, opposite direction to phase speed

As f0, Kelvin wave degenerates to plane gravity wave; Generated by ocean tides and wind near coastal areas

y

x

u=0

0

y

vH

t

yg

t

vx

gfv

Rx

Rx

ectyHF

ectyFgHv

u

/

/

)(

)(

0

gHc

f

cR ,

32

Natural frequency of a stratified system

A simple 1D model for continuously stratified fluid Horizontally homogenous: (z) Static equilibrium initially Wave motion due to buoyancy

The freq. of oscillation is

If N2 > 0, stable stratification internal wave motion possible If N2 < 0, unstable stratification small disturbances leads to

overturning; no internal waves possible

z

(z+h)

(z)

(z-h)

0

)()()(

02

2

2

2

hdz

dg

dz

hd

zhzgVdz

hdVz

dz

dgN

0

2 Brunt-Vaisala frequency

33Internal waves Surface & internal gravity waves: same origin (interface); interchange

of kinetic and potential energy Quintessential internal waves:

Over an infinite domain (even in the vertical!) with vertical stratification No ambient rotation (gravity waves) Neglect viscosity; small-amplitude motion Hydrodynamic effects are important

Perturbation expansion:

Wave solution ~ exp[i(lx+my+nzt)]

gz

ppptxpzpp

txz

0~

0~

0

1 |,||'| ),,(' )(

|||'| ),,(')(

(u,v,w), generated by perturbation, are small 0'

:or ,0'

0

''1

'1

'1

02

00

0

0

wg

N

tdz

dw

t

z

w

y

v

x

u

gy

p

t

w

y

p

t

v

x

p

t

u

Transport eq.

2 22 2

2 2 2 l m

Nl m n

34Properties of internal waves For a given N, maxN Dispersion relation only depends on the angle between the wave number and

horizontal plane: For a given frequency, all waves propagate at a a fixed angle to the horizontal As 0, 90o

Wave motion:

Transverse wave Group vel. inside the phase plane Purely vertical motion when n=0; Cg=0 no energy radiation Purely horizontal motion when l=0; =0 steady state; blocking For continuously stratified fluid on a finite depth, the phase speed is bounded by a

maximum but there are infinitely many modes of internal waves Inertial-internal waves:

cosN

)

0),(

)cos('

)sin(

~

220

22

wu

tnzlxUn

nl

g

N

un

lw

tnzlxUunl

lN

Nf

)

C

x

zCg