The momentum equations for a rotating shallow homogeneous fluid with a free upper surface (i.e. the shallow water equations) may be written as Əh Du Dt Dv Dt = +fv-g Əh ду = -fu-g- (1) where u(x, y, t), v(x, y, t) are the velocities on the free surface, and h(x, y, t) is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator D Ə Ә Ə +u- +v Dt Ət əx ду du = (5 + 1) = − (5 + 1) (² Dt (2) (b) Use the two equations you developed in Part(a) to derive the evolution equation for $+f, where = Qu. In other words, show that dy Əv + əx ду (3) being careful to explain the steps. [Hint: start by deriving the equation for and then use the fact that the Coriolis parameter f = f(y) only.]

The momentum equations for a rotating shallow homogeneous fluid with a free upper surface (i.e. the shallow water equations) may be written as Əh Du Dt Dv Dt = +fv-g Əh ду = -fu-g- (1) where u(x, y, t), v(x, y, t) are the velocities on the free surface, and h(x, y, t) is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator D Ə Ә Ə +u- +v Dt Ət əx ду du = (5 + 1) = − (5 + 1) (² Dt (2) (b) Use the two equations you developed in Part(a) to derive the evolution equation for $+f, where = Qu. In other words, show that dy Əv + əx ду (3) being careful to explain the steps. [Hint: start by deriving the equation for and then use the fact that the Coriolis parameter f = f(y) only.]

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.

Question

The momentum equations for a rotating shallow homogeneous fluid with a free upper
surface (i.e. the shallow water equations) may be written as
Du
Dt
Dv
Dt
Əh
дх
Əh
ду
= +fv-g
for (+ f, where =
=
=
-fu-g-
where u(x, y, t), v(x, y, t) are the velocities on the free surface, and h(x, y, t) is the depth
of the free surface.
(a) Rewrite (1) and (2) by expanding the total derivative operator
D Ə
Ə
= +2= +v
Dt Ət əx ду
(b) Use the two equations you developed in Part(a) to derive the evolution equation
Əv
- Oy. In other words, show that
dy
(1)
(2)
?u Əv
+
дх ду
{ (S + ƒ ) = − ( 5 + £ ) ( ²
Dt
(3)
being careful to explain the steps. [Hint: start by deriving the equation for and
then use the fact that the Coriolis parameter f = f(y) only.]

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Explain the physical meaning of each of the terms in Equation 3. The momentum equations for a rotating shallow homogeneous fluid with a free upper surface (i.e. the shallow water equations) are equation 1 & 2. where u(x, y,t), v(x, y,t) are the velocities on the free surface, and h(x, y,t) is the depthof the free surface.