Transcript Document

Lecture 7: Finite-Volume Methods
Unlike finite-difference and finite-element methods, the computational domain in the finite-volume
methods is divided into many control volumes (CV) and the governing equations are solved in its
integral form in individual control volumes.
For example:
 [




1
   (v D)]dxdy  0 
   vn Dds
t
t
s
(7.1)
Structured grids
1. Assign the elevation at the center
of each rectangular control volume;
N
vD
W
uD
uD
vD
S
E
2. Define that outflow is positive
and inflow is negative;
3. Calculate the net flux
Approximation of volume integrals
un1 D
un D

y
x
uD
 x dxdy  S uDdy  [un Dn  un1 Dn1 ]y
uD
xy  [u n Dn  u n 1 Dn 1 ]y
x
uD [u n Dn  u n 1 Dn 1 ]

x
x
n-1
un1 D
n
n+1
un D
un1 D
x
The first order upwind scheme
u n 1 Dn 1  u n Dn u n Dn  u n 1 Dn 1
uD
dxdy

uDdy

[

]y
 x
S
2
2

y
(u n 1 Dn 1  u n 1 Dn 1 )
2
uD
y
xy 
(u n 1 Dn 1  u n 1 Dn 1 )
x
2
uD u n 1 Dn 1  u n 1 Dn 1

x
2x
The second order central scheme
Consider an arbitrary function like
F   fds
On the east side, for the first order approximation,
Fe  f e y
nw
n
ne
For the second order approximation,
f
w
sw
e
P
s
y
Fe 
se
1
( f se  f ne )y
2
For the fourth order approximation,
x
Fe 
Consider an arbitrary function like
F   fd  fxy

y
( f se  4 f e  f ne )
6
For the first order approximation
F  f P xy
For the second order approximation,
F  fxy
The fourth order approximation can be obtained by using the bi-quadratic shape funcion:
f ( x, y)  ao  a1 x  a2 y  a3 x 2  a4 y 2  a5 xy  a6 x 2 y  a7 xy 2  a8 x 2 y 2
Need 9 coefficients,which can determined by fitting the function to the value of f at 9
locations (nw,w,sw, n, p,s, ne,e,and se).
F  xy[ao 

a3
a
a
(x) 2  4 (y) 2  8 (x) 2 (y ) 2 ]
12
12
144
xy
(16 f p  4 f s  4 f n  4 f w  4 f e  f se  f sw  f ne  f nw )
36
For the cell-centered grids, the value at P point is known,but values at other points must be
obtained by interpolation from surrounding cell-centered nodes.
Comments;
Structured grid finite-volume model is a special type of the finite-difference methods and they
always can convert from one to another. So, little efforts need to make to convert a finitedifference model to a finite-volume model under structured grids.
3. Popular unstructured triangular FVM grid in CFD:
1.
Cell-centered
2.
Cell-vertex overlapping
3.
Cell-vertex median
Characteristics of the oceanic motion:
●
Free surface----How to calculate accurately the integral form of the surface pressure gradient
forcing?
●
Steep bottom topography----How to ensure the mass conservation in a two mode model
system?
●
Open boundary conditions----How to minimize the wave energy reflection at open
boundaries?
Cell vertex median grid
A Grid: All variables
(,,u,v, , , s..) at nodes
Cell-centered
B Grid: All variables
(,,u,v, , , s..) at centroids
Cell-vertex-centered
,
u , v , u, v
,  , s, Km, Kh…
C Grid:
Advantage:
Advantage:
Advantage:
1) Simple
2) Guarantee the mass
conservation for tracers
1) Simple
2) Better to advection
calculation

Combine the best of A and
B Grids;

Easy to ensure the mass
conservation for tracers

Easy to introduce the mass
conservative open boundary
conditions
Disadvantage:
Disadvantage:
The accuracy of the surface
elevation gradient forcing is
sensitive to the shape of the
control element (due to
interpolation)
Hard to guarantee The
accuracy of the surface
elevation gradient forcing
Hard to ensure the mass
conservation
at
open
boundaries
Hard to ensure the mass
conservation
at
open
boundaries
Hard to ensure the mass
conservation
for
tracer
calculation
FVCOM
FVCOM (Unstructured Grid, Finite-Volume Coastal Ocean Model)
•
Hydrostatic and free surface primitive equation ocean models;
•
Mellor and Yamada turbulence model for vertical mixing and Smagorinsky eddy
parameterization for horizontal mixing, recently upgraded to include general ocean turbulence
models (GOTM).
•
Sigma transformation in the vertical direction, and unstructured triangular
horizontal plane;
•
Two mode approaches: barotropic 2D mode and baroclinic 3D mode; with second order
approximation;
•
Exact form of the no flux boundary conditions on the slope;
•
Flooding/drying processes;
•
Multi-choices of the radiation open boundary conditions;
•
NPZ, multi-species NPZD, sediment suspension model and water quality model.
grids in the
The -transformation coordinates:

z 
z 

H 
D
The governing equations in the -transformation coordinate:
 Du Dv 



0
t
x
y 
uD u 2 D uvD u
 gD 
D 1 
u



 fvD   gD

[ ( D   d )  
]
(Km
)  DFx
t
x
y

x o x 
x
D 

0
vD uvD v 2 D v
 gD 
D 1 
v



 fuD   gD

[ ( D   d )  
]
(Km
)  DFy
t
x
y

y o y 
y
D 

0
D  uD  vD  1 





(K h
)  DHˆ  DF
t
x
y

D 

sD s uD s vD s 1 
s




(K h
)  DFs
t
x
y

D 

The boundary conditions:
(
u v
D
,
)
( sx , sy );
 
o K m
(
 = 0;
u v
D
,
)
( bx , by ); ,  = 0;
 
o K m

D

[Qn ( x, y, t )  SW ( x, y,0, t )];
 c p K h

AH D tan 

;
 K h  AH tan2  n
s
s( P  E ) D


Kh
s
AH D tan s

 K h  AH tan2  n
at  = 0
at  = -1
Two-mode (external and internal) approach:
The 2D equations: solve for 
The 3D equations: solve for u, v, w, , s, Km, Kh
The 2-D (vertically integrated) equations:
 u D  v D 


0
t
x
y
 
u D u 2 D u v D
 gD 
D
~


 fv D   gD

{ ( D   d )d 
  d }  sx bx  DFx  Gx

t
x
y
x o 1 x 
x 1
o
0
0
0
0
0
0
 
v D u v D v 2 D
 gD 
D
~


 fu D   gD

{ ( D   d )d 
  d }  sy by  DFy  Gy

t
x
y
y o 1 y 
y 1
o
Gx  (
u 2 D u v D
u 2 D uvD
~

)(

)  DFx  DFx
x
y
x
y
Gy  (
u v D v 2 D
uvD v 2 D
~

)(

)  DFy  DFy
x
y
x
y

u

u v
~
DFx  [2 Am H
]  [ Am H (  )]
x
x
y
y x
DFx 

u 
u v
2 Am H

Am H (  )
x
x y
y x

u v

v
~
DFy  [ Am H (  )]  [2 Am H ]
x
y x
y
y
DFy 

u v

v
Am H (  )  2 Am H
x
y x y
y
Prr
Turbulence Closure Submodels
1. Horizontal diffusion coefficient: A Smagorinsky eddy parameterization method
a) for momentum:
Am  0.5Cu (
u 2
v u
v
)  0.5(  ) 2  ( ) 2
x
x y
y
where C : a constant parameter; u: the area of the individual momentum control element
b) for tracers:
Ah 
0.5C
Pr
(
u 2
v u
v
)  0.5(  ) 2  ( ) 2
x
x y
y
where  : the area of the individual tracer control element;
Pr:
the Prandtl number.
The surface and bottom boundary conditions for temperature are:

z

z
z  ( x , y ,t )

z  H
z
1

[Qn ( x, y, t )  SW ( x, y,  , t )]
c p K h
n
0
AH tan 
Kh
n
SW ( x, y, z, t )  SW ( x, y,  , t ) [Re
z 
a
 (1  R)e
z 
b
]
The absorption of downward irradiance is included in the
temperature (heat) equation in the form of

SW ( x, y, z, t ) SW ( x, y,0, t ) R a 1  R b
Hˆ ( x, y, z, t ) 

[ e 
e ]
z
 cp
a
b
z
z
The surface and bottom boundary conditions for salinity are:
s
z
s
z

s( Pˆ  Eˆ )
Kh 

AH tan s
K h n
z  ( x , y ,t )
z  ( x , y ,t )
ˆ : Evaporatio
Pˆ : Precipitation rate;E
n rate
The kinematic and heat and salt flux conditions on the solid boundary:
v n  0;

s
 0;
0
n
n
l

0
z
 AH tan 

z
K h n
The 2-D finite-volume discrete approach:
u,v
1. The surface elevation:
ζ
 t dxdy   [
u D  v D 

]dxdy   vn Dds 
x
x
s
2. Velocity:
ui ( x, y)  iu ( x, y)  ui ,0  aiu x  biu y
u,v
u,v

u,v
u,v
u,v
y
vi ( x, y)  iv ( x, y)  vi ,0  aiv x '  biv y '
aiu , biu , aiv , and biv :
determined by a least-square method based on
velocity values at the four cell centered points
The second-order accuracy upwind scheme
(Kobayashi, 1999) to calculate the momentum and
volume fluxes
x
Time integration: The modified fourth-order Runge-Kutta time-stepping scheme. This is a multistage time-stepping approach with second-order accuracy.
ζ
u D  v D 
dxdy


[
 t
 x  x ]dxdy  s (u Ddy  v Ddx)  R
u D

gD2 
  D

dxdy


u
D
v
d
s

f
v
Ddxdy

gD
dxdy

{
[

d


 t
s n 
 x
 o 1 x 
  D x d ]d }dxdy
0
 
0
0
 sx   bx
~
dxdy  DFx dxdy  Gx dxdy  Ru
o
v D

gD2 
  D
 t dxdy  s v Dvn ds   fu Ddxdy  gD y dxdy  { o 1[ y  d    D y d ]d }dxdy
0
 
 x
NT ( j )
n
i
0
i
n
i
0
i
 

;
k
uin1  ui4 ; vin1  vi4
k =1,2,3,4
ζ nj1  ζ 4j
j :
tRu0
u  u 
;
4ui Di
k
i

v  y 2m1u mn D2nm1  x2 m vmn  y 2 m u mn D2nm
n
2 m 1 m
( 1, 2 , 3 , 4 )  1/ 4,1/ 3,1/ 2,1
; and
2 j
u u ,v v
0
i
m 1
tRk 1

0
 sy   by
~
dxdy   DFy dxdy  G y dxdy  Rv
o
ζ0j  ζ nj ; R0  Rn 
ζ kj  ζ 0j   k
0
R R , R R
0
u
n
u
0
v
n
v
tRv0
v  v 
4 iv Di
k
i
0
i
k
The area of the tracer element (TCE)
ui and iv : The area of individual triangle
(momentum element) (MCE)
The 3-D Internal Mode:
Two step:
1.
Use either explicit scheme or second-order accuracy Runge-Kutta time stepping scheme
to calculate “immediate” velocities by solving the momentum equations
uD
  Ru (advection, Coriolis, pressure gradient, surface elevation and horizontal diffusion terms)
t
vD
  Rv (advection, Coriolis, pressure gradient, surface elevation and horizontal diffusion terms)
t
2. Use the implicit scheme to solve the equation including vertical diffusion terms
uD
1 
u
 Ru 
(K m
)
t
D 

vD
1 
v
 Rv 
(K m
)
t
D 

The implicit method used to calculate the vertical diffusion is the same as POM/ECOM-si
The calculation of the tracer advection terms:
s
s
a) The second-order upwind scheme:
s 1

sdy;
x  b 
s 1

sdx
y  b 
s
s
u,v
s
s
s  so  x  y
x
y
(The current version of the FVCOM code uses this approximation)
s
u,v
u,v
s
s
b) The first-order central scheme ??
s
u,v
u,v
u,v
3
s 
s
k 1
i
s
s
3
(The very early version of the FVCOM code uses this approximation)
c) The working-on scheme:
Keep the upwind scheme as (a) but using the least square to determine the tracer distribution in the big
area with inclusion of the central point. This could allow a highly order accuracy scheme.
Transport Consistency of External and Internal Modes:
tE
tI
I split 
t I
t E
Internal mode:
I split
i ,k 1  i ,k 
 k n1
 k n
( i   in ) 
v N ,k Ddl
t I
 i l
To conserve the volume on the ith TCE, the vertically-integrated form of the above equation must satisfy
 in1   in
t I
 k n
v Ddl  0
n  N ,k

k 1
i
l
kb1

Since the sea level is calculated through the vertically integrated continuity equation over Isplit external time
steps, then the above condition requires
split
 k n
1
t I  n  v N ,k Ddl  t E  nˆ  v Nnˆ Ddl
k 1  i l
n1  i l
kb1
I
Mass Conservative Open Boundary Conditions:
, ,
Case 1: The sea level at the open boundary is specified (tidal elevation)
Step 1:
 u
FC 
  ( FA  FB )
t

, ,
Step 2: Determine the tracer flux by
 FC
FA

Step 3: Apply a radiation condition for the perturbation of the tracer
such as
FC
FB
, ,
    

Case 2: The sea level at the open boundary is unknown
,, 
Step 1: Apply a radiation condition to determine the sea level at the open boundary
Step 2: Repeat steps 1-3 listed in Case 1
Recommended shapes of triangular grids at open boundaries:
Advantage:
, 
, 
u,v
Avoid the numerical error due to interpolation
u,v
, 
, 
u,v
u,v
Note: For a region in which the topography
varies abruptly at the open boundary, a sponge
layer is always recommended in these area after
the above treatment.
, 
, 
u,v
u,v
, 
, 
Methods to Add the Discharge from the Coast or a River :
Method 1: The TCE Method---Inject the water into the tracer control element (TCE)

 [ v n Dds  Q] /  
t
s

Q is the water volume transport into a TCE with an
area of   and a depth of D
The vertically averaged
caused by Q equals to
Uo 
ˆ
Q cos ˆ
D(li  l j )
Vo 
S
u,v
velocity
Q sin ˆ
D(l i  l j )
S
u,v
u,v
i
S
j
S
S

k
o
is the angle of the coastline relative to the x direction.
For the external mode:
The contributions of Q to the x and y vertically-integrated momentum equation in the MCE with an area of  i or  j are given as
0.5QVo
0.5QU o
For the internal mode:
KM 1
the percentage of Q in the kth sigma layers which satisfy the condition of
Ris
Qk
U ok 
QRQk cos ˆ
D(li  l j ) k
; Vok 
QRQk sinˆ
D(li  l j ) k
0.5QRQk U ok
0.5QRQk Vok
R
k 1
Qk
1
Methods to Calculate the Salinity at the Freshwater Discharge Node Point
1) Specified: The value of the salinity is specified by users.
2) Calculated: The value of the salinity at the freshwater discharge node point is calculated
using the salinity equation as follows:
Sok D
S

 [  vnk Sk Dds   Fs dxdy QRQk Sˆok ] /  
( K h ok )
t
D

s ( li l j )

v nk is the velocity component normal to the boundary of the TCE in the kth sigma layer
S k is the salinity at nodes of triangles connecting to the coastal node of the TCE
Sˆ ok
is the salinity contained in the water volume of Q
Fs
is the horizontal and vertical diffusion terms in the salinity equation
Method 2: The MCE Method ---Inject the water through the boundary edge
The vertically averaged velocity caused by Q:
Uo 
Q
cos ˆ
Dl
Vo 
Q
sin ˆ
Dl
The x and y components of the velocity
in the kth sigma layer in the MCE are
given as
U ok 
QR Qk cos ˆ
Dl  k
; Vok 
S
S
So
The contributions to the internal x and y
momentum equations of the MCE in the kth
sigma layer are equal to
QRQk U ok
QRQk Vok
The transport inject to the TCEs that include the river edge:
Qi  Q j  Q / 2
And then, the sea level at the node point is calculated by
 I
 [ vnI DI ds  QI ] / I
t
s
S
u, v
u, v
QR Qk sin ˆ
Dl  k
u, v
S
u, v
So
u, v
S

Criterions for horizontal resolution or time step:
Considering a case with no vertical and horizontal diffusion, the vertical integral form of the salinity equation
with a point freshwater discharge can be written as
SD
 [ v n SDds  QSˆ o ] / 
t

Let us start with a simple first-order time integration scheme:
S
n 1
n
D
 S ( n1 
D
n

t vn D n ds
D
n 1


)
tQSˆ o
D n1
Including the continuity equation, yield:
S n 1  S n 
tQ
ˆ  Sn)
(
S
o
D n 1 
For a freshwater discharge case,
S n 1  (1 
Sˆo  0
tQ
)S n
n 1 
D 
For mass conservation,
tQ
1  n 1   0
D 
L~
I splitQ
D gD
or
D n 1
t 
Q
For example:
Q ~ 10 3 m3/s, D ~ 10 m, and I split  10 , L ~ 100 m
The wet/dry point treatment:
 D  H    hB

 H  0; hB  0
Criterion for node points:
z

hB
0
H
D  H    hB  Dmin : wet node;
D  H    hB  Dmin : dry node
Criterion for triangular cells:
D  min(hB,i , hB, j , hB,k )  max( i ,  j ,  k )  Dmin : wet cell;
D  min(hB,i , hB, j , hB,k )  max( i ,  j ,  k )  Dmin : dry cell
Mass conservative!

u D v D
dxdy


[
 t
 x  y ]dxdy  S vn Dds
, S
u, v
Critical issues:
How to ensure the mass conservation of the individual TCE in which the moving boundary
is crossed?
During the Isplit external mode time steps, due to drying of nodes or triangles,

n 1
i

I split
n
i
1
nˆ
1  i
n
 t E 
v
nˆ
N
Ddl
if Isplit =1
l
In this case, the external and internal mode adjustment can not guarantee that  reaches
zero at =-1.
• An additional adjustment of  is added in the  equation
Note: This adjustment works in general but fails in the case where =-Dmin<10-1
The key parameters controlling the mass conservation:
• Criterion to define the wet/dry points;
• Time step used for numerical integration;
• Horizontal and vertical resolutions of model grids;
• Amplitudes of surface elevation;
• Bathymetry.
QS. How could we estimate these constraints?
Conduct a sensitivity study using the model with an simple geometric river
with inclusion of intertidal zones.
Wind Stress, Heat Flux, Precipitation/Evaporation and Tidal Forcing
a) Wind stress: 1) Calculated directly in the model (time and spatial dependent or uniform)
2) Input directly from the output of MM5 or other meteorological models
b) Heat flux: 1) Specified by users (net flux and short-wave radiation)
2) Input directly from the output of MM5 or other meteorological models
c) Precipitation/Evaporation: The code has never been used!
d) Tidal forcing:
No
 o   o  ˆi cos(i t   i )
i 1
N o the total number of tidal constituent
 o the mean elevation relative to a water level at rest
Six tidal constituents are
ˆ i the amplitude of the ith tidal constituent
1)
2)
3)
4)
5)
6)
i the frequency of the ith tidal constituent
 i the phase of the ith tidal constituent
S2 tide (period = 12 hours);
M2 tide (period = 12.42 hours);
N2 tide (period = 12.66 hours);
K1 tide (period = 23.94 hours);
P1 tide (period = 24.06 hours),
O1 tide (period = 25.82 hours).
The Data Assimilation Methods
1)
Nudging method: Directly adopted from the weather forecast model (MM5)
Simple, computational efficiency and practical to the forecast application
1.
2.
3.
Sea surface SST
Current mooring data at fixed locations
Hydrographic data in the 3-D domain
2) The OI method: This method will be added into the Fortran 90 code this summer
Simple, statistical meaningful, computational efficiency and practical to the forecast
application, but requires covariance data.
3) The adjoint method: This method will be added by collaborating with Carl Wunsch at MIT
Better to sensitive studies, but requires huge computational times for iteration.
The 3-D Lagrangian Particle Tracking
FVCOM has two online (run with the code) and offline Lagrangian particle tracking programs
by solving a nonlinear system of ordinary differential equations (ODE) as follows

dx  
 v ( x (t ), t )
dt
This equation is solved by using 4th order explicit Runge-Kutta (ERK) multistep methods
1.
An individual-based zooplankton and fish larvae model has been developed on this
Lagrangian frame.
2.
Vertical and horizontal random walk processes have been included (Chen et al.,
2004)
The Nutrient-Phytoplankton-Zooplankton Model:
P
Grazing
Z
Mortality
Egestion
Uptake
N
Figure 8.1: Schematic of the food web loop of the NPZ model
The NPZDB Model
Figure 8.2: Schematic of the lower trophic level food web model for a
phosphorus-controlled ecosystem system.
The NPZD Model
Figure 8.3: Schematic of the flow chart of the multi-species NPZD
model.
The Water Quality Model
Boundary
Flux
Water Column
NO3+NO2
OPO4
Uptake
Uptake
Death
Death
Death
OP
Settling
Reaeration
SOD
DO
Bacterial
Respiration
Settling
Photosynthesis
Respiration
PYHT
Benthic Flux
Death
ON
Settling
Denitrification
Oxidation
Settling
Nitrification
NH3
Mineralization
CBOD
Light
Temperature
& Salinity
Mineralization
Advection
& Dispersion
Denitrification
External
Loading
Sediment
Schematic of the water quality model in the sediment layer.
Vertical Diffusion
Denitrification
CBOD
Oxidation
DO
Phytoplankton
Death
NH3
ON
Mineralization
OP
OPO4
Mineralization
NO3+NO2
Sediment Anaerobic Layer
The On-going Improvement
1. Add GOTM modules into the current FVCOM version 2 to provide multiple choices for
vertical mixing ;
2. Add the OI data assimilation method (which we had for the old version) to the current
FVCOM version 2
3. Add the high-order advection scheme for tracer equations
4. Test the off-line biological models
The future Improvement
1.
Convert FVCOM to the non-hydrostatic version
2.
Add adjoint data assimilation to FVCOM for both online and offline
3.
Convert SWAN to unstructured grids and then couple into FVCOM
The Final Goal
Build a atmospheric-ocean coupled community model system.
Model Validation
1. Advection Scheme
2. Wind-induced oscillation;
3. Wind-induced waves over the slope bottom topography;
4. Tidal Resonances in semi-enclosed channel;
5. Freshwater discharge plume;
6. Bottom boundary layer over a step bottom slope
7. Equatorial Rossby soliton
8. Flooding/drying process
9. Dye Experiments
1. Advection Scheme
F
F
C
0
t
x

5,
F ( x,0)  
 0,
2 x  2
otherwise
and C = 1
Upwind finitedifference stream
Central finitedifference stream
FVCOM finite-volume
flux scheme
Upwind finitedifference stream
Central finitedifference stream
FVCOM finitevolume flux scheme
Wind-induced oscillation
Wind is suddenly imposed at
initial
Linear, non-dimensional equations:
Wind
u

 v  
t
r
ro=67.5 km
v

 u  
t
r

  ( ru )
v

[

]0
t
r
r

gd
g
   r cos
,     ,   o 4 ,  o  3 4

ro f
ro f
 0 , (u, v, ) r 0  finite, u t 0  v t 0  0 , 
d=75 m
where  
and
u r 1
t 0

 (r, )
Solution:

o
 (r , , t )  4 [ Ao (r ) cos   a k Ak (r ) cos(   k t )]

k 1

 o Ao (r )
u (r , , t )  3 [(
 1) sin    bk Fk (r ) sin(   k t )]
r

k 1

 o dAo (r )
v(r , , t )  3 [(
 1) cos   bk Gk (r ) cos(   k t )]
dr

k 1
Reference:
Csanady ( 1968)
Birchfield (1969)
FVCOM
POM
Water elevation
Alongshore transport
Radial mode: k=1, 2: gravity waves, k=3: topographic wave
Birchfield and Hickie (1977) JPO
Analytical
FVCOM (5 km)
POM (2.5 km)
1h
1d
5 days
: Elevation

V : Current
FVCOM (5 km)
POM (2.5 km)
Tidal Resonance in A Rectangular Channel
z

 u

g
0
 t
x

u H
 

0


t

x

u x  L1  0,  x  L  A co st


B.W
x
B
L1
L
Solution:
  [C1 J o (2k x )  C 2Yo (2k x )]cost
k 2   2 L / gH o
  2 / T  2 / 12.42(hour)
1) Normal case:
H o  20 m, L  290 km;
H L  10 m, B  16 km
1
O.B.
2) Near-resonance case:
H o  20 m, L  290 km;
H L  0.67 m, B  16 km
1
Ho
x  2.5 km
Normal case:
FVCOM
Computed
Analytical
POM
ECOM-si
Near-resonance case
FVCOM
POM
ECOM-si
Freshwater discharge
Case 1: Continental shelf
Case 2: Circular lake shelf
Z
400
Y
300
100m
m
X
10
0
Sea Level
Salinity
FVCOM
FVCOM
POM
POM
Central difference scheme
Upward difference scheme
x
(km)
4.2
2.0
0.9
The Bottom Boundary Layer
Background mixing coefficient Km = 10-4 m2/s
Slope:

300 m
 0.02
3
13.5  10 m
Bottom boundary condition:
T
0
z
FVCOM
x  5 km
POM
10 days
u(cm/s)
Interval: 0.1
w(10-3 cm/s)
Interval: 0.2
FVCOM
u (cm/s) POM
x  2.25 km
5 days
10 days
AH D tan T
T

 K h  A H tan2  n
Vertical velocity is one order of magnitude smaller!
Interval: 0.05
U
(cm/s)
W
(cm/s)
Equatorial Rossby Soliton
Non-dimensional grid size (scaled by the wave length)
x=0.125
Model-calculated
full solution
Zero-order analytical
solution
A simple estuarine model used to test the model sensitivity
30 km
   o cos(t )
T  2 / 
 12.42 hours

10 m
3 km
2.3 km
FVCOM: Two time steps: TE: External mode; TI: Internal mode
= 410-4
= 710-4
= 910-4
Figure 3.10: The model-predicted relationship of the ratio of the internal to external mode time
steps () with the tidal forcing amplitude () and the bathymetric slope of the inter-tidal zone (). In
these experiments, = 4.14 sec, kb = 6, = 5 cm.
= 410-4
= 710-4
= 910-4
Figure 3.11: The model-derived relationship of Isplit
with Dmin for the three cases with = 4.010-4, 7.010-4
and 9.010-4. In the three cases, = 4.14 sec, kb = 6,
and = 1.5 m.
kb=11
kb=6
Figure 3.12: The model-derived relationship of
Isplit with Dmin for the two cases with kb = 6
and 11, respectively. In these two cases, = 4.14
sec, = 4.010-4, and = 1.5 m.
L = 600 m
L = 300 m
tE=2.07 sec
tE=4.14 sec
Figure 3.13: The model-derived relationship of with (upper panel) and (lower
panel). In the upper panel case, = 4.14 sec, = 7.010-4, kb =6, and = 1.5 m. In the
lower panel case, = 300 m, = 7.010-4, kb =6, and = 1.5 m.
The 1999 Dye Experiment (carried by Bob Houghton)
4km grid
2km grid
`
At 16: 33 GMT, 22 May, 1999
0.5 km grid
0.25 km grid
At 16: 33 GMT, 22 May, 1999
1: 3:59 GMT, 23 May
2: 11: 43 GMT, 23 May
3: 15:22 GMT, 23 May
4: 19:24 GMT, 23 May
5: 21:34 GMT, 23 May
6: 00:53 GMT, 25 May
6
5
2
1
3
4
Observed
Simulated
Assimilation with T and S
Simulated 0.5 km
Assimilation 0.5 km
Trajectory of a nearsurface particle
Trajectory of a nearbottom particle
Trajectory
of
the
center of the vertically
averaged model dye
concentration.
0
1
2
3
Lapse times (days)
4
5
End here! More model validations will be given by Dr.Wang in the next lectures