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Modified PIC Method for Sea Ice Dynamics
WANG Ruixue ( 王瑞学)a, JI Shunying ( 季顺迎)a, b, 1,
SHEN Hungtaoband YUE Qianjin ( 岳前进)a
aState Key Laboratory of Structural Analysis for Industrial Equipment,
Dalian Universi ty of Technology , Dalian 116023, China
bDepartment of Civil and Environmental Engineering, Clarkson University, Potsdam, NY 13699, USA
( Received 31 January 2005 accepted 10 June 20 05)
ABSTRACT
The sea ice cover displays various dynamical characteristics such as breakup, rafting , and ridging under external
forcesTo model the ice dynam ic process accurately, the effective numerical modeling method shou ld be establishedIn
this paper, a modified particleincell ( PIC) method for sea ice dynamics is developed coupling the finite difference
( FD) method and smoothed particle hydrodynam ics ( SPH ) In this method, the i ce cover is first discretized into a series
of Lagrangian ice particles which have their ow n sizes, thicknesses, concentrations and velocities The ice thickness and
concentration at Eulerian grid positions are obtai ned by interpolation w ith the Gaussian function from their surrounding ice
particlesThe mom entum of ice cover is solved w ith FD approach to obtain the Eulerian cell velocity, which is used to
estimat e the ice particle velocity with the Gaussian function also The thickness and concentration of ice particles are ad
justed with particle mass density and sm ooth length, which are adjusted w ith the redistribution of ice particlesWith the
above modified PIC method, numeri cal simulations for ice motion i n an idealized rectangular basin and the ice dynamics
in the Bohai Sea are carried out These simulations show that this modified PIC method is applicable to sea ice dynamics
simulation
Key words particleincell smoothed particle hydrodynamics sea i ce dynamics numerical simulation
This study w as financially supported by the National Natural S cien ce Foundation of China ( Grant No4020 6004)
1 Corresponding author Emailjisy dlut educn
1Introduction
Under natural conditions, the sea ice is a mixture of rafted and ridged ice, level ice and open wa
ter, and displays various dynamic properties A series of numerical methods have been developed to
simulate the sea ice dynamical process under different scales Normally, the ice cover is treated as a
two dimensional rheological flow, and the Eulerian finite difference ( FD) method is applied widely to
the polar zone at large scale, and the subpolar area at mesoscale ( Hibler, 1979 Lepparante and Hi
bler, 1985 Lu et a l ,1989 Zhang and Hibler , 1997 Wu et a l ,1998) In the solving of advec
tion in the continuity equation by the FD method, the numerical diffusion gives poor prediction of ice
edge position ( Huang and Savage, 1998) Some researchers developed other Lagrangian methods to
overcome the numerical diffusion problem ( Shen et al ,2000 Tremblay and Mysak, 1997 Overland
et al ,1998 Wang and Ikeda, 2004)
The Discrete Element Model ( DEM) was adopted for sea ice dynamics under Lagrangian coordi
nate at different scales ( Shen et al ,1987 Dai et al ,2004 Hopkins, 1996 Hopkins et al ,
China Ocean Engineering , Vol 19, No 3 , pp457468
2005 China Ocean Press, ISSN 08905487
2004) The DEM is a good approach to the study of the mechanisms of ice floe interaction, icewave
interaction, ice rafting and ridging, but it has not been used for general ice forecasting and long term
simulation for its low computational efficiency As another Lagrangian approach to sea ice dynamics,
the Smoothed Particle Hydrodynamics ( SPH) was also applied successfully in the riversea ice dynam
ics ( Gutfraind and Savage, 1997a, 1997b S hen et al ,2000 Lindsay and Stern, 2004) The SPH
method can simulate the ice edge accurately, avoiding the numerical diffusion p roblem Meanwhile,
the huge interpolating calculation among particles, especially in the determination of ice strain rate,
stress and internal ice force, is very costly Thus, the computational efficiency of SPH method should
be improved for sea ice forecasting and simulation
The particleincell ( PIC) method is a suitable one with a coupling of Eulerian and Lagrangian
coordinates Flato ( 1993) firstly introduced the PIC method into the sea ice dynamics for the polar
zone, and it was found that the PIC method had an obvious advantage in determining the accurate ice
edge position In the PIC method for sea ice simulation, a bilinear function was adopted to interpolate
the ice variables between ice particles and grid nodes In the previous PIC method, the ice particle did
not have concentration, and its area was a feedback variable of the grid concentration, w hich was ad
justed with the critical concentration condition ( A10) at grid node ( Flato, 1993) Moreover, the
concentration of grid node could also be solved with its continuity equation, while the numerical diffu
sion still appeared in the solution of the continuity equation with the FD method ( Huang and Savage,
1998)
In this paper, the PIC method is modified with flexible particles and a Gaussian interpolating
function based on the SPH theory The Gaussian function has smoother continuity and higher precision
than the bilinear function The ice concentration and thickness of ice particle can be determined with
its mass density and smooth length in ice motion In this modified PIC method, the ice velocity at grid
node is determined by solving the momentum equation with FD method It is faster than the SPH
method which has complex iterations in solving strain rate, stress and internal ice force With this
modified PIC method, the ice ridging process in a rectangular zone, and the ice dynamic process in the
Bohai Sea are tested In the two numerical tests, the simulated ice results are in good agreement with
the analytical solution and observed data for the Bohai Sea
2Basic Equations of Sea Ice Dynamics
21Momentum Equation
The momentum equation for the motion of the ice field is governed by ice interactions, wind and
water forces, the Coriolis force, and ocean tilt effect, and can be written as
MdV
dtMfKV
a
wMg
w ( h) ( 1)
where, Mis the ice mass per unit area, and M
ih, the mean ice thickness hNhi,
ibeing the
ice density, hibeing the ice thickness, Nbeing the ice concentration Vis the ice velocity vector f
is the Coriolis parameter Kis a unit vector normal to ice surface
aand
ware the air and water
458 WANG Ruixue et a l China Ocean Engineering , 19(3) , 2005,457 468
stresses, and here
a
aCaVai Vai , and
w
wCwVwi Vwi , in which
aand
ware the den
sities of air and current, Caand Cware the drag coefficients of wind and current, and Vai and Vwi are
the relative wind and current velocity vectors gis the acceleration of gravity
wis the ice surface
height is the ice stress vector
22ViscousPlastic Constitutive Model
In the simulation of sea ice dynamics, the ice cover is normally treated as a 2D continuous medi
um, and the viscousplastic ( VP) constitutive model with elliptical yield curve law is applied most
widely ( Hibler, 1979) This VP law can be expressed as
ij 2
ij ()
kk
ij P
ij2(2)
where
ij and
ij are the 2D stress and strain rate tensor Pis the ice pressure
ij is the Kronecker
operator and are the nonlinear bulk and shear viscosities, and can be determined as
(
ij , P ) min( P2,
0), (3)
e2, ( 4)
in which
2
1
2
11e2, and here
1
11
22 and
n(
11
22)24
2
12 eis the eccen
tricity of the elliptical yield curve Based on Eqs ( 3) and ( 4) , we have
0and
0e2when
the ice cover has a small shear rate ( P2
0)Under this condition, the ice cover exhibits linear
viscous rehological charactertics When the ice cover has a large shear rate ( P2
0) , the ice
principal stresses lie on the ellipitical yield curve, and the ice cover displays plastic rehology
The pressure term is calculated by ( Shen et al ,1990)
Ptan2
4
21
i
w
ighi
2
N
Nmax
j
(5)
where is the internal friction angle of surface ice, Nmax is the maximum possible ice concentration,
and jis an empirical constant set to 15 normally The and signs are for passive and active
states of ice flow
3Modified PIC Method for Sea Ice Dynamics
In this modified PIC method, the ice cover is discretized into a series of ice particles, which have
their own locations, velocities, thicknesses, concentrations and sizes With the Gaussian kernel func
tion, the ice information at a grid node can be interpolated from their surrounding ice particles The
ice velocity at a grid node is solved with FD method, and used to determine the ice particle velocity
With redistribution of ice particles in the Lagrangian system, the mass density of ice particles w ill be
adjusted to determine the particle thickness and the concentration of the ice particles
31Ice Thickness and Concentration at Eulerian Grid Node
The sea ice thickness and concentration at the Eulerian grid nodes can be interpolated with the
Gaussian kernel function from their neighboring sea ice particles Based on the SPH theory ( Shen et
al, 2000) , a field variable fat position rcan be expressed as
459
WANG Ruixue et al China Ocean Engineering , 19(3) , 2005,457 468
f ( r)
N
k1
mk
Mkf( rk)W(rrk, h0), (6)
where ris the position vector for estimating variable f , rkis the position vector of ice particle k , Mk
and mkare mass density and mass of particle k, and h0is the smooth length, which determines the
range of influence of the interpolation kernel In this study, the Gaussian kernel foundation is adopt
ed, w hich has good continuity and precision as the interpolating function The spatial distribution of
the 2D G aussian kernel function is plotted in Fig 1, and can be written as
W( rrk, h0)1
h2
0
exp (rrk)2
h2
0
(7)
Based on Eq ( 6) , the ice thickness and concentration at grid node Xi, j can be determined from
its neighboring ice particles by
hi(Xi, j )
N
k1
mk
MkW( Xi, j rk, h0) hi(rk) , ( 8)
Ni(Xi, j )
N
k1
mk
MkW( Xi, j rk, h0)N(rk) , ( 9)
where Xi, j is the position vector of the grid node hi(rk)and N ( rk)are the thickness and concentra
tion of ice particle khi(Xi, j )and N ( Xi, j)are the estimated thickness and concentration at grid
node Xi , j
Fig1Sketch of spatial distribution
of 2D Gaussian function
32Sea Ice Velocity at Eulerian Grid Node
The ice velocity at a grid node can be calculated from the sea ice momentum equation with Eulerian
FD method The components in xand ydirections of the sea ice momentum equation can be written as
U
tUU
xVU
yfV g
w
x(
ax
wxFx)M
V
tUV
xVV
yfU g
w
y(
ay
wyFy)M
(10)
where, Uand Vare ice velocities in xand ydirections
axand
ay,
wxand
wyare the compo
nents of air and water drag stress Fxand Fyare internal ice force components, which can be deter
mined by
Fx
x(
xxh)
y(
xyh) , Fy
y(
yy h)
x(
y xh) , ( 11)
460 WANG Ruixue et a l China Ocean Engineering , 19(3) , 2005,457 468
in which,
xx ,
yy ,
xy and
y x are the ice stress components Based on the viscous plastic constitutive
model of Hibler ( 1979) , the sea ice stresses can be written as
xx
xx
xx
yy
yy P
2
yy
yy
xx
yy
xx P
2
xy
y x 2
xy
(12)
where,
xx ,
xy and
yy are strain rate components
In the present study, the FD method has a spatial central difference scheme with a staggered grid
and a three level time difference scheme ( Fig 2) In the first half time step from nt( n 12)
t, the velocity component in xdirection, Un12
i12, j, is solved with an implicit scheme, and the ve
locity component in ydirection, Vn12
i , j 12, is solved with an explicit scheme In the second half time
step from ( n 12)t( n 1)t, the velocities, V
n1
i, j 12and Un1
i12, j , are solved with implicit
and explicit schemes in yand xdirections, respectively
for M, A, h , H, and P
for ,, and
ij
for Ua,Uw, and U
fo r Va,Vw, and V
Uaand Uw,Vaa nd Vware the wind and current v e
locities in xand ydirections, r espectively
Fig2Coordinate of difference cell for sea ice momentum equation
33Velocity and Position of Sea Ice Particle
With the ice velocity ( Ui12, j , Vi, j12) at a grid node solved with FD method at time tnor
tn12, the velocity vector of ice particle k, V( rk( tn) ) or (U(rk),V( rk))can be estimated from
the velocities at its neighboring grid nodes, and can be written as
U ( rk)
i
j
mi, j
Mi, j W( Xi , j rk, h0) Ui12, j (13)
V( rk)
i
j
mi, j
Mi, j W( Xi , j rk, h0) Vi , j12(14)
where, U(rk) and V(rk) are the velocity components in xand ydirections of ice particle k, respec
tively Ui12, j and Vi, j 12are the sea ice velocity components at the grid node mi, j and Mi, j are
the mass and mass density of grid h0is the smooth length Here we have h01
2(xy ) , xand
ybeing the grid size in xand ydirections, respectively
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WANG Ruixue et al China Ocean Engineering , 19(3) , 2005,457 468
The velocity vector of sea ice particle kat time tn12or tn1can be determined with Eqs ( 13)
and ( 14) If its location vector at time tnis rk( tn), then its position vector rk( tn12)at time tn12
can be calculated with
rk( tn12)rk( tn)t
2V( rk( tn)), (15)
where tis the time step
34Thickness and Concentration of Sea Ice Particles
In a previous PIC study for sea ice dynamics, Flato ( 1993) interpolated the ice thickness and
concentration at grid nodes from the volume and area of ice particles And the ice particle area was ad
justed based on the feedback of the estimated con centration in cells with the critical condition Nmax
10Huang and Savage ( 1998) determined the physical ice thickness in cells with its continuity equa
tion by FD method, then determined the mean ice thickness in cells w ith interpolation from ice particle
thicknessesThe ice concentration at the cell centers was evaluated with NhhpIn the present
study, the mass density and smooth length of ice particles are introduced for the calculation of thick
ness and concentration of particles Then, the thickness and concentration in cells can be interpolated
from the thickness and concentration of particles
If the ice particle khas position vector rk,its mass density can be estimated from its neighboring
particles,
M( rk)
N
j1
mjW( rkrj, h 0)(16)
The smooth length h0has a close relationship with mass density If the initial smooth length is
h(0)
0, its smooth length at nth time step can be determined with
h( n)
0h(0)
0
M(0)
M( n)
1
2(17)
where h(0)
0and h( n)
0are the initial smooth length and the smooth length at time step nM(0)and
M( n) are the initial mass density and the mass density at time step n
The mass density of particle kcan also be written as M ( rk)
iN( rk) hi(rk) , and its concen
tration can be determined by
N( rk)M( rk)
ihi(rk)(18)
If the simulated ice concentration N( rk)10, ice ridging or rafting will occur In this situa
tion, we set N ( rk)Nmax 10, and have the ice thickness hi(rk)M( rk)(
iNmax)
With the above modified PIC method, the sea ice dynamics can be simulated in the first half time
step from tnto tn12or the second half time step from tn12to tn1In this method, a Gaussian
kernel function is used to interpolate the ice variables between Lagrangian particles and Eulerian grids
The thickness and concentration of ice particles are determined with their smooth length and mass den
sity With the modified PIC method, the sea ice dynamics could be modeled without numerical diffu
462 WANG Ruixue et a l China Ocean Engineering , 19(3) , 2005,457 468
sion Its computational cost is lower than that of the SPH method for its does not have the complex in
terpolation in the calculation of ice particle interactions
4Numerical Simulation of Ice Ridging in A Regular Domain
Under constant wind and current drags, the ice ridging process in a rectangular domain is simulat
ed for verification of the modified PIC method The initial ice condition is that ice of thickness ti0 and
concentration Ni0 is distributed uniformly over a rectangular region of length Land width B, as show n
in Fig 3Under constant wind and current drags, the ice cover will pile up at the downstream end
The internal ice resistance increases with the ice thickness to balance the wind and current d rag forces
In the steady state, the ice strain rate
ij 0, and the ice concentration approaches its maximum val
ue, i eNmax 10In this simulation, the gravitational gradient and the Coriolis effect are both ig
nored
Fig3Sketch of the initial distribution of sea
ice over the regular region
With the smooth boundary, the analytical solution for the thickness profile of the static ice ridge
can be obtained as ( Shen et al ,2000)
tit2
i0 2(
aCaV2
a
wCwV2
w)
tan2
4
21
i
w
ig
x ( 19)
where ti0 is the single layer ice thickness at the ice edge, and xis the distance from the leading edge
to the ice ridge, where titi 0
In this numerical test of ice ridging, the initial ice region LB20 km 20 km, the initial
thickness ti0 02 m, and the initial concentration N0100 The current velocity is set at zero,
the wind velocity is 15 ms, and the wind direction is 270In the simulation, the time step t9
0 s, the grid size xy400 m 400 m, and there are 2 2 particles in a cell initially
The wind and current drags are the main driving forces in the sea ice dynamics, and the drag co
efficients are the most important in their calculation The wind and current drag coefficients vary with
different sea ice conditions Based on the results for different regions and the sea ice characteristics in
the Bohai S ea ( Ji et al ,2003) , we adopt the wind coefficient Ca00015 and the current coeffi
cient Cw00045, respectively The main parameters are listed in Table 1
463
WANG Ruixue et al China Ocean Engineering , 19(3) , 2005,457 468
Table 1 Parameters used in t he ice ridging simulation
Parameter Value Parameter Value
Ice frictional angle 46Bulk viscosity
010106Nsm2
Air density
a129 kgm3Shear viscosity
025105Nsm2
Wind drag coefficient Ca00015 Current drag coefficient Cw00 045
Water density
w1010 kgm3Ice density
i910 kgm3
With the modified PIC method, the simulated widthaveraged thickness is plotted in Fig 4It
can be seen that, under the given wind in the 270direction, the ice ridging approaches the steady
state in 14 hours, and the mean ice thickness is consistent w ith the analytical solution The measured
results for the Bohai Sea show that the level ice thickness is generally smaller than 0 2 m, and the
rafted ice thickness can be over 0 4 m In general, the thickness of the hummocked ice is 1 2 m,
and its maximum value can be 3 5 m ( Wu et al ,2001) The ridge height in this sample is 1 31
m, which is reasonable based on measured data for the Bohai Sea
Fig4Compari son of simulated result with analytical solution in static ice ridge thickness profile
For the wind direction of 225, the simulated ice thickness contour and the ice velocity vector are
shown in Fig 5It can be found that, under the wind drag, the ice cover ridges at the downwind cor
ner of the domain In 20 hours, the ice ridge approaches a steady state
5Simulation of Sea Ice in the Bohai Sea
To examine the validity of the modified PIC method, we simulate the sea ice dynamics of the
Liaodong Bay for 72 hours from 13 40, Jan 22, 2004, and compare the results with the satellite re
mote sensing image and field data The initial ice thickness and concentration are obtained from the
digital NOAA remote sensing image, and displayed on the first picture in Figs 6 and 7, respectively
The wind velocity measured at Jz202 platform ( 12121, 4030) is used and the wind field in the
464 WANG Ruixue et a l China Ocean Engineering , 19(3) , 2005,457 468
Fig5Distributions of simulated ice velocity and ice thick ness
Liaoding Bay is assumed to be uniformThe tidal current is calculated with 2D shallow water equation
by ADIFD algorithm In the simulation, the time step is 600 s, the grid size is 0 101, and
there are 5 5 particles in one cell initially The other computational parameters are listed in Table 1
51Simulated Ice Distribution in the Liaodong Bay
The ice thickness contours simulated with the modified PIC method for different times are plotted
in Fig 6In this figure, the satellite remote sensing images are also given for comparison with the
simulated results It can be seen that the modified PIC can simulate the ice dynamics wellThe simu
lated ice concentrations are mostly in the range from 80 to 100 , which can be found from Fig 7
Within the 72 hours of numerical simulation, the simulated ice edge drifted to the south area under the
north wind action, and the simulated ice concentration was decreased slightly If the ice thermodynam
ics is considered in the model, the simulated results should be more reasonable
52Simulated Results for Jz202 Area of Liaodong Bay
The sea ice parameters for the Jz202 area of the Liaodong Bay are interpolated from its neighbor
ing particles with the Gaussian function The simulated ice thickness and velocity in the 72 hours are
465
WANG Ruixue et al China Ocean Engineering , 19(3) , 2005,457 468
Fig6Simulated ice thickness contour and satellite rem ote sensi ng image at different times
Fig 7Distributions of sea ice concentration simulated at different time steps
plotted in Fig 8The ice information observed on the JZ202 oilgas platform is also plotted in this
466 WANG Ruixue et a l China Ocean Engineering , 19(3) , 2005,457 468
Fig 8Simulated and measured i ce inform ation for JZ202 area in 72 hours
figure It can be found that the observed ice thickness lies in the range from 9 cm to 13 cm, which is
in agreement with the simulated data Under the strong tidal current, the ice velocity displays a regular
fluctuation, which can be observed from the simulated data The simulated velocity is in good agree
ment with the field measured data
6Conclusion
In the numerical simulation of sea ice dynamics, it is very important to develop effective and ac
curate numerical methods In this study, a modified particleincell ( PIC) method is developed cou
pling the Eulerian FD method and Lagrangian SPH model In this modified PIC method, the Gaussian
kernel function is adopted instead of the bilinear function, and the mass density and the smooth length
of ice particles are used to determine the particle thickness and concentration In this way, the numeri
cal diffusion of the Eulerian FD method can be avoided, and the Gaussian function with perfect conti
nuity is more precise than the bilinear function The ice velocity at the Eulerian grid is simulated with
finite difference method, and the movement of ice is determined under Lagrangian coordinate The ice
parameters are transferred between the Eulerian gird and Lagrangian ice floe by means of Gaussian in
terpolation By use of the modified PIC method, the ice ridging process in a rectangular region and sea
ice dynamics of the Bohai Sea are simulated, and the results are in good agreement with the analytical
solution and observed data The computation efficiency of the modified PIC is evidently higher than
that of SPH For instance, in the case of ice ridging, the computation time with the modified PIC is 15
467
WANG Ruixue et al China Ocean Engineering , 19(3) , 2005,457 468
minutes, while it is 50 minutes with SPH The reason is that with the modified PIC, the sea ice stress
and motion are calculated based on the Eulerian FD method, thus, the time step is much larger than
that for the SPH calculation under Lagrangian coordinate Therefore, both the computation efficiency
and precision of the modified PIC method are high in the numerical simulation of sea ice dynamics
AcknowledgementsThe authors appreciate the study of REU student Katrina Ligett from Brown University , USA, and
the helpful discussions with ProfH ayley Shen and DrLianwu Liu f rom Clarkson University, USA
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