Question

1.) Continuity Equation (3 Pkt.) Demonstrate that it is possible to write the continuity equation in the form dt Show that this is equal to the form: dt Derive a diagnostic equation for the vertical wind component w(z) with the assumption of large scale movements.

Answer 1

From the Eulerian deviation of Continuity equation rightarrow rho u - partial differential/partial differential x (rho u) delta x/2 (in) out = rho u + partial differential/partial differential x (rho u) delta x/2 Total = [rho u - partial differential/partial differential x (rho u) delta x/2] delta y delta z - [rho u - partial differential/partial differential x (rho u) delta x/2] delta y delta z = -partial differential/partial differential x (rho u) delta x delta y delta z - (1) Similarly = -partial differential/partial differential y (rho v) delta x delta y delta z - (2) (z) = -partial differential/partial differential z (rho w) delta x delta y delta z - (3) Man inflow per unit volume (1) + (2) + (3)/delta x delta y delta z = -nabla^rightarrow . (rho u^rightarrow) u^rightarrow = u x + v y+ w z Which should be equal to partial differential rho/partial differential t \r\n Rightarrow partial differential rho/partial differential t = -nabla . (rho u^rightarrow) partial differential rho/partial differential t + nabla . (rho u^rightarrow) = 0 - (1) We can convert this into a total derivative (or) By using the vector identity