A Semi-Implicit, Three-Dimensional Model for Estuarine ... - USGS

2. Governing Equations and Boundary Conditions 19
This expression describes the rate of change in density of a given mass of fluid as it is followed through the flow. Using the
substantial derivative, equation 2.1 can be rewritten as
1
ρ˜ -- Dρ˜ ∂ũ
------i
+ ------- = 0 . (2.3)
Dt ∂x
i
Because the compressibility of water is small, it is usually assumed thatDρ˜ ⁄ Dt = 0 for water flows of constant density
everywhere, so that equation 2.3 can be simplified to
∂ũ
i
------- = 0 . (2.4)
∂x
i
When the estuarine water density is a function of salinity, temperature, and pressure, it is not correct to equate Dρ˜ ⁄ Dt to zero
simply by invoking incompressibility. The formal definition of incompressibility requires only that the fluid density not be affected
by changes in pressure, which is not equivalent to the statement that Dρ˜ ⁄ Dt = 0 if the density of the moving fluid element is
changed by the molecular conduction of heat or the exchange of water molecules for salt ions. However, for estuaries, these effects
are miniscule and can be safely ignored when considering mass or volume conservation. Therefore, equation 2.4 can in fact be used
as a good approximation to the instantaneous equation for mass conservation in estuaries.
2.2.2 Momentum Conservation
The principle of conservation of linear momentum arises from Newton’s second law of motion and requires that the change
of momentum in a control volume fixed to the rotating reference frame be equal to the net influx of momentum into the control
volume and the sum of the forces acting on it. In the form of a differential equation, this principle results in the following instantaneous
equation of motion for estuaries (Pritchard, 1971a):
where
∂ρ˜ ( ũ
i
) ∂ρ˜ ( ũ
--------------i
ũ
j
)
∂
+ ---------------------
∂p˜
– ------- – 2ε
∂t ∂x
j
∂x ijk
Ω
j
ρ˜ ũ
k
ρ˜ g
i
ρ˜ ν
i
2ũ i
= + + --------------- , (2.5)
∂x
j
∂x
j
g i is the local acceleration of gravity in the direction x i ,
p˜ is the instantaneous pressure,
Ω j is the component of the angular velocity of the earth in the direction x j,
ν is the coefficient of (molecular) kinematic viscosity (assumed constant), and
εijk is the alternating tensor defined by the relations ε123 = ε231= ε312 = 1, ε213 = ε132 = ε321 = –1, and εijk = 0 if two or more
subscripts are equal.
Equation 2.5 is the famous Navier-Stokes equation in tensor notation including gravitational and Coriolis body-force terms; the
tensor equation represents three scalar equations corresponding to the three possible values of the free subscript i.