Chapter III
Numerical Modeling
34
The speeds of the currents at the average depth are defined as follows:
Equation III-16
Equation III-17
The lateral stress forces í µí± í µí±í µí± consist of viscous friction, differential advection, and turbulent
friction. These forces are computed based on turbulent viscosity using the formulation of average
velocity gradients at depth.
Equation III-18
Equation III-19
Equation III-20
Table III-4 Symbols and Significations in Shallow Water Equations and Current Velocities.
Symbol
Signification
Symbol
Signification
t
Time.
A
Horizontal viscosity.
Pa
Atmospheric pressure.
g
Gravity acceleration.
x. y et z Cartesian coordinates.
í µí½ í µí²
Water density.
η
Surface elevation.
í µí±º í µí²í µí² ,í µí±º í µí²í µí² í µí±º í µí²í µí²
et í µí±º í µí²í µí²
Component of the radial
tensor.
d
Water depth.
í µí½ í µí²
Turbulent vertical viscosity.
h = η+d Total water depth.
í µí½ í µí¿
The reference density of
water.
u, v et w
Velocity components in the x, y,
and z directions.
S
Amplitude of discharge
caused by point sources.
(í µí² í µí² ,í µí² í µí² )
Velocity at which water is
discharged into the ambient water.
(í µí½ í µí²í µí² ,í µí½ í µí²í µí² )
x and y components of bottom
wind stress.
(í µí½ í µí²í µí² , í µí½ í µí²í µí² )
x and y components of surface
wind stress.
Tij
Lateral stresses.
u, v
Velocities of currents at the mean
depth.
f = 2Ω sinΦ
Coriolis parameter {Ω is the
angular velocity and Φ is the
geographical latitude}.
Ω
Angular velocity of rotation.
Numerical Modeling
34
The speeds of the currents at the average depth are defined as follows:
Equation III-16
Equation III-17
The lateral stress forces í µí± í µí±í µí± consist of viscous friction, differential advection, and turbulent
friction. These forces are computed based on turbulent viscosity using the formulation of average
velocity gradients at depth.
Equation III-18
Equation III-19
Equation III-20
Table III-4 Symbols and Significations in Shallow Water Equations and Current Velocities.
Symbol
Signification
Symbol
Signification
t
Time.
A
Horizontal viscosity.
Pa
Atmospheric pressure.
g
Gravity acceleration.
x. y et z Cartesian coordinates.
í µí½ í µí²
Water density.
η
Surface elevation.
í µí±º í µí²í µí² ,í µí±º í µí²í µí² í µí±º í µí²í µí²
et í µí±º í µí²í µí²
Component of the radial
tensor.
d
Water depth.
í µí½ í µí²
Turbulent vertical viscosity.
h = η+d Total water depth.
í µí½ í µí¿
The reference density of
water.
u, v et w
Velocity components in the x, y,
and z directions.
S
Amplitude of discharge
caused by point sources.
(í µí² í µí² ,í µí² í µí² )
Velocity at which water is
discharged into the ambient water.
(í µí½ í µí²í µí² ,í µí½ í µí²í µí² )
x and y components of bottom
wind stress.
(í µí½ í µí²í µí² , í µí½ í µí²í µí² )
x and y components of surface
wind stress.
Tij
Lateral stresses.
u, v
Velocities of currents at the mean
depth.
f = 2Ω sinΦ
Coriolis parameter {Ω is the
angular velocity and Φ is the
geographical latitude}.
Ω
Angular velocity of rotation.