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