Chapter 8 Configuring Fluidity - The Applied Modelling and ...

20 Model equations
where y is the local coordinate in the Northwards direction. Taking ϕ = ϕ0 + y/RE and expanding
(2.34) in a Taylor series yields
f = f0 + 2Ω cos ϕ0
y
RE
where RE ≈ 6378 km. Typical values of these terms are:
Ω =
+ . . . , β = 2Ω
cos ϕ0,
RE
2π
24 × 60 × 60 = 7.2722 × 10−5 rad s −1 ,
(NB. a sidereal day should be used to give a more accurate value of 7.2921 × 10 −5 rad s −1 ) and
2.4.2 Linear Momentum
ϕ0 = 30 ϕ0 = 45 ϕ0 = 60
f0 7.2722e-05 1.0284e-04 1.2596e-04
β 1.9750e-11 1.6124e-11 1.1402e-11
Newton’s second law states that the sum of forces applied to a body is equal to the time derivative
of linear momentum of the body,
� f = d(mu)/dt. (2.38)
Applying this law to a control volume4 of fluid and making use of equation 2.9 leads to the linear
momentum equation for a fluid which can be written as
�
�
� �
∂
ρudV = − uρu · ndA + F ρdV + σ · ndA, (2.39)
∂t
S
V
S
V
where V is the control volume, S is the surface of the control volume and n is the unit normal to the
surface of the control volume and F is a volume force per unit mass. Physically, 2.39 states that the
sum of all forces applied on the control volume is equal to the sum of the rate of change of momentum
inside the control volume and the net flux of momentum through the control surface. More details
regarding the derivation and properties of this equation can be found in Batchelor [1967].
2.4.3 Buoyancy and Hydrostacy
For fluids upon which gravity is acting the buoyancy force should be considered when a free surface
is present or when the fluid contains variations in density. The buoyancy force is given by b = −ρg
and, in the absence of viscosity, a simple form of the vertical momentum equation can be written as
Dw
Dt
where b = ρg is the magnitude of the buoyancy force.
If the fluid is in a state of rest, the left hand side of (2.40) is zero giving
= −∂p + b, (2.40)
∂z
∂p
∂z
= b. (2.41)
This is known as hydrostatic balance. It states that the pressure at point is equal to the weight of
water above that point, plus any pressure loading on the surface of the domain (e.g., atmospheric
pressure or ice load which is here termed pa):
�
p(x) = pa + b.
4 The concept of a control volume and how they are defined within fluidity is discussed more in section 3.2.4
z