General Curvilinear Ocean Model (GCOM): Enabling ...

General Curvilinear Ocean Model
(GCOM): Enabling Thermodynamics

M. Abouali, C. Torres, R. Walls, G. Larrazabal, M. Stramska,
D. Decchis, and J.E. Castillo
AP0901

09

General Curvilinear Ocean Model (GCOM): Enabling
Thermodynamics

M. Abouali∗,1, C. Torres, R. Walls, G. Larrazabal, M. Stramska, D. Decchis1, and J.E. Castillo1.
1San Diego State University
∗Corresponding author: [email protected]

Abstract: General Curvilinear Ocean helps to distribute the energy from those re-
Model (GCOM) is a coastal ocean model gions of the earth, which receive more en-
curvilinear in 3 dimensions, developed by ergy, such as the equator, to those regions
Carlos Torres et al. The model uses a di- where solar energy budget is much less; thus,
rect numerical simulation (DNS) approach keeping the weather from being too cold or
to solve the primitive Navier-Stokes equa- too warm. It is well known that the thermo-
tions and uses boundary fitted Curvilinear haline circulation in the ocean had affected
coordinates; therefore, it is possible to use the ice age in the past.
GCOM in various topographies and meshes.
GCOM is currently being coupled with heat It is no further needed to emphasize the
and salinity conservation (or thermohaline importance and how strongly the ocean is
dynamics) equations. GCOM is designed affecting us, the human, and other animals.
in a modular fashion, which will make it There are couple of methods used to study
possible to easily add biogeochemical sub- the ocean and its impact which can be cat-
models to study different phenomenon, such egorized as: (1) Observational, (2) Analyti-
as phytoplankton blooms, transport of pol- cal, (3) Laboratory, and (4) Numerical stud-
lution, or biogeochemical cycling. In a sep- ies. The focus of this paper is on the latest,
arate project an Atmosphere-Ocean Interac- i.e. numerical studies and numerical models.
tion model is developed, which will be cou-
pled later to GCOM. In this paper we introduce the General
Curvilinear Ocean Model, abbreviated as
Keywords: Ocean Model, General Curvi- GCOM, which was originally developed by
linear Coordinate, Navier-stokes, Thermo- Prof. C. Torres and Prof. J.E. Castillo
dynamics [6] [5]. GCOM uses full 3D general curvi-
linear coordinate system. This feature en-
1 Introduction ables the model to adapt very well to the
real physical boundary of the study area
About two third of the Earth is covered with both in vertical and horizontal; unlike the
oceans and seas. Beside the complex ecosys- sigma coordinate which is designed to adapt
tem existing inside the ocean and being on just to the bathymetry; hence, the verti-
of the main sources of food for human be- cal direction. GCOM currently uses non-
ing, the ocean is interacting with the Earth dimensional primitive Navier-Stokes equa-
Atmosphere and controlling its properties. tions with Boussinesq approximation and it
Hence, indirectly affecting our other sources was previously used in many projects.
on lands. Furthermore, the oceans are one of
the main sources of solid particles required In next section, we will explain the main
in the formation of the clouds. equations building the core of the GCOM
model. Later, we explain the thermody-
The huge specific heat capacity of the namic equations, which are under process of
ocean has made it a big natural heat and being added to the model. As the curvilinear
energy storage. During the summer, while equations are being used, we need to transfer
there exist excess of solar energy, the heat both the grid and the underlying equations
energy is absorbed and stored in the ocean into a rectangular uniform grid to ease the
and during the winter this excess of energy is implementation in the computer and apply-
released and helps to have a milder weather. ing the boundary conditions [1] [2]. There-
Not just throughout the year, the ocean also fore, we explain the necessary steps and re-
lations needed for this transformation in 3
dimensional spaces. At the end, the future

of the model is explained. 2.2 Continuity Equation

2 Governing Equations The continuity is given as:

In this section we introduce the core equa- ∂ρ + ρ · ∇u = 0 (3)
tions used in GCOM, i.e. Navier-Stokes ∂t
(NS) equations [7]. we focus on both dimen-
sional and non-dimensional form of the NS
equations [4] [3].

2.1 Momentum equation 2.3 Poisson Equation

The dimensional form of the Navier-Stokes The Navier-Stokes equation does not provide
equation is: any relation for the pressure. Thus, we need
to find a method to calculate the pressure.
∂u + u · ∇u = −gkˆ − 2ω × u − 1 ∇p + ν∇2u One of the commonly used method is to ap-
∂t ρ ply a divergence operator on the momentum
equation, 2, and then by the use of conti-
(1) nuity equation, 3, derive a simplified equa-
where u is the velocity, g is the acceler- tion for the pressure. Although we used the
ation due to gravity, ω is the coriolis, ρ0 is term simplified, but it does not mean that
the density, ν is the viscosity, and p is the the the resulting equation is easy to solve.
pressure. Each term has its unique meaning. As the matter of fact, the resulting equa-
from left to right: tion is called the poisson equation for the
pressure and is one of the expensive part of
• Term I: is the storage of the momen- the numerical model to solve. The resulting
tum. poisson equation for pressure is given in the
4. It has to be noted that this equation is
• Term II: is the advection. mostly due to numerical purposes and does
not have a physical meaning.
• Term III: is the Effect of the gravity/

• Term IV: is the effect of Coriolis ∇2p = − 1 ∇ · ρkˆ − ∇ · [(u · ∇) u ]
Forces, due to the rotation of the Fr2
earth.
+ 1 ∇2 D − ∂D + 1 ∇ · vˆi − uˆj
• Term V: is the pressure gradient Re ∂t Ro
forces.
(4)
• Term VI: is the influence of the vis-
cous stresses. where we have:

Using the Reynolds number, i.e. Re = D = ∇·u (5)

VL , Froude Number, i.e. Fr = √V ,
ν
gL

and Rossby Number, i.e. Ro = V the
Lf

above equation can be written in its non- 3 Transformation

dimensional form as follow:

∂u +u·∇u = − ρ + 1 vˆi − uˆj −∇p+ 1 ∇2u As mentioned earlier, GCOM uses curvilin-
∂t Fr2kˆ Ro Re ear coordinate system. Therefore, it is pos-
sible to use GCOM in the variety of the
(2) bathymetry and grids. But both the grids
and the equation are needed to be trans-
where V is the reference velocity and L formed to a rectangular uniform grid as in
1.
is the reference length, chosen by the user to

non-dimensionalize the equations.

a = ξx2 + ξy2 + ξz2 (10)
b = ηx2 + ηy2 + ηz2
c = ζx2 + ζy2 + ζz2
d = ξxηx + ξyηy + ξzηz
e = ζxηx + ζyηy + ζzηz
q = ξxζx + ξyζy + ξzζz

4 Thermodynamic Equations

Figure 1: Transforming the grid The original GCOM code was not coupled
with the thermodynamic equations; thus,
Once the grid is transformed we also need its usage was limited to the neutral condi-
to transform the equations. As we have just tion. As the development, the thermody-
transformed the grid, this means that just namics equations are being coupled with the
the spatial derivative are needed to be trans- model. In equation, the amount of potential
formed and there is no need to transform temperature and salinity are two important
the time derivative. The transformed spa- factor. These two scalar variable, together
tial derivative can be easily obtained using with pressure define what the density is in
the chain-rule in derivation as follow: the certain location of the ocean. The equa-
tion which relates the pressure, salinity, and
the potential temperature to the density is
called Equation of State:

∂ = ξx ∂ + ηx ∂ + ζ x ∂ ρ = f (p, θ, S) (11)
∂x ∂ξ ∂η ∂ζ

∂ = ξy ∂ + ηy ∂ + ζ y ∂ (6)
∂y ∂ξ ∂η ∂ζ

∂ = ξz ∂ + ηz ∂ + ζ z ∂ 4.1 Conservation of Heat
∂z ∂ξ ∂η ∂ζ

where ξx, ηx, · · · are called the metrix The heat equation is given by:
of the transformation and can be obtained
using: ∂θ + u · ∇θ = νθ ∇2 θ + Qθ (12)
∂t

xξ xη xζ ξx ξy ξz (7) where νθ is the thermal diffusivity and
yξ yη yζ · ηx ηy ηz = I Qθ is the sink or source term and it is invari-
zξ zη zζ ζx ζy ζz ant to coordinate transformation. Different
model has to be executed to account for in-
The three dimensional transformed coming solar radiation, amount of evapora-
Laplacian operator is: tion, or any other interactions between the
ocean and the atmosphere.

∇2(f ) = L(f ) 4.2 Salinity Equation or
Conservation of Heat
−L(x) [ξx ∂(f ) + ηx ∂(f ) + ζ x ∂(f ) ]
∂ξ ∂η ∂ζ The conservation of Salinity can be written
as:
−L(y) [ξy ∂(f ) + ηy ∂(f ) + ζ y ∂(f ) ]
∂ξ ∂η ∂ζ

−L(z) [ξz ∂(f ) + ηz ∂(f ) + ζ z ∂(f ) ]
∂ξ ∂η ∂ζ

(8)

where we have: ∂§ + u · ∇S = νS ∇2S + QS (13)
∂t

L(f ) = a ∂ 2 (f ) + b ∂ 2 (f ) + c ∂ 2 (f ) + where νS is the salinity diffusivity and QS is
∂ξ2 ∂η2 ∂ζ2 the sink or source term and it is invariant to
∂2(f ) ∂2(f ) ∂2(f ) coordinate transformation. As an example,
2 [d ∂ξ∂η + e ∂ζ∂η + d ∂ξ∂ζ ] if we have precipitation, which is of different
salinity than the ocean, It must be take care
(9) of in this term.

and the coefficients in 9 are defined as

follow:

5 Solution Method is also being changed to the mimetic method
developed by Prof. Castillo. This method
In previous sections different equations that replaces the divergence and the gradient op-
are used in GCOM were discussed. But the erator with its equivalent discrete operator.
order that these equations are executed is as
follow: Currently the code solves the given equa-
tion directly. This requires a very fine mesh
• Step 1: First all fields are initialized at the Kolmogorov scale. For high Reynolds
and the model parameters are read number and real application, this requires
from a text file. a very fine mesh; hence, a huge amount of
memory and computation time, which is not
• Step 2: The Poisson equation for the available even on the most powerful exist-
pressure is solved and the new pressure ing supercomputers, and it is believed that
field is obtained. this amount of memory will not be available
in near future either. Therefore, a closure
• Step 3: The momentum equation is model, or a sub-grid model is required to ac-
solved and the new velocity field is ob- count for the energy and the effect of small
tained. scale eddies. This step is being currently
studied and added to the code.
• Step 4: Using the updated velocity
field the Thermodynamic equations, References
both conservation of heat and conser-
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putational fluid dynamics for engineers,
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state a new density field is recalcu-
lated. [2] P. Knupp and S. Steinberg, Fundamen-
tals of grid generation, ISBN:0-8493-
• Step 6: Different fields are prepared 8987-9.
for the next iteration. Required fields
for post processing are written to text [3] J. Pedlosky, Geophysical fluid dynam-
file. ics, , ISBN:0-387-96388-X , ISBN:0-387-
96388-X ISBN:0-387-96388-X.
• Step 7 The time is advanced by dtI
and f the desired time interval has not [4] R.B. Stull, An introduction to boundary
reached, we continue on step 2. layer meteorology, ISBN:90-277-2769-4.

6 Future of the GCOM [5] C.R. Torres and J.E. Castillo, A
new 3d curvilinear coordinates numer-
GCOM was originally developed in FOR- ical model for oceanic flow over arbi-
TRAN 77 in a single file. It is more desired trary bathymetry, Desarrollos Recientes
to change the code in to a modular fash- en Metodos Numericos (2002), 105–112.
ion, by using different subroutines. The code
is also rewritten in FORTRAN 90 using its [6] , Stratified rotating flow over
parallel and vector operation capabilities. complex terrain, Applied Numerical
Mathematics 47 (2003), 531–541.
Furthermore, the GCOM is currently us-
ing second order central finite difference. It [7] C.R. Torres, H. Hanazaki, J. Ochoa, J.E.
has been shown that higher order mimetic Castillo, and van Woert M.L., Flow past
orders behave much better in such a highly a sphere moving vertically in a stratified
non-linear and complex system of equations. diffusive fluid, Journal of Fluid Mechan-
Therefore, the numerical scheme of the code ics 417 (2000), 211–236.