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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 Əh (1) Dt +fv-9ax Əh Du = -fu- gay (2) Dt where u, v are the velocities on the free surface, and h is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator ə D Ә ə = +u+v Dt Ət dz dy (b) Use the two equations you derived in Part(a) to derive the evolution equation for S+f, where =-. In other words, show that D 1/1₁ (5 + ƒ) = − (5 + ƒ) ( Dt Əu Əv + Әх 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 Du Əh (1) Dt +fv-9ax Əh Du = -fu- gay (2) Dt where u, v are the velocities on the free surface, and h is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator ə D Ә ə = +u+v Dt Ət dz dy (b) Use the two equations you derived in Part(a) to derive the evolution equation for S+f, where =-. In other words, show that D 1/1₁ (5 + ƒ) = − (5 + ƒ) ( Dt Əu Əv + Әх 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
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Hint: take partial derivatives of the given shallow water equations and subtract them, e.g.  d(eq2)/dx  -  d(eq1)/dy, and look for bits that make up vorticity, vorticity advection and divergence

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 Əh +fv-g (1) Dt дх Du Əh -fu- gJy (2) Dt where u, v are the velocities on the free surface, and h is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator Ə D Ə Ə = + u Dt Ət əx dy S+f, where C = = = (b) Use the two equations you derived in Part(a) to derive the evolution equation for - In other words, show that dy +v= ?u ; (6 + A) = − (S + A) (3) Dt -OH), (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.] + дх ду
Transcribed Image Text: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 Əh +fv-g (1) Dt дх Du Əh -fu- gJy (2) Dt where u, v are the velocities on the free surface, and h is the depth of the free surface. (a) Rewrite (1) and (2) by expanding the total derivative operator Ə D Ə Ə = + u Dt Ət əx dy S+f, where C = = = (b) Use the two equations you derived in Part(a) to derive the evolution equation for - In other words, show that dy +v= ?u ; (6 + A) = − (S + A) (3) Dt -OH), (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.] + дх ду
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