Show that if the wind is blowing parallel to the isotherms that the temperature advection is zero.

Perform a scale analysis of the horizontal momentum equations for the vortex formed in your sink while draining by using typical scales given below. Which terms are important in this case? $U$~~$0\mathrm{.}01\frac{m}{s},W$~~$0\mathrm{.}01\frac{m}{s},L$~~$0\mathrm{.}01m,H$~~$0\mathrm{.}1m,p$~~$0\mathrm{.}1Pa,f$~~${10}^{-4}{s}^{-1},v={1\mathrm{.}810}^{-6}\frac{m^2}{s},a={10}^{7}m,$~~$1000k\frac{g}{m^3}.$

Assume that the gravity force at the surface is $g_0=9\mathrm{.}80665\frac{m}{s^2}.$ Calculate the geopotential height at an altitude of $5000$ meters for a$)$ Gravity is constant with height and b$)$ Gravity decreases with height.

For the following scalar fields find the magnitude of the gradient at the point indicated.

a$.a(x,y,z)=2x^3{y}^2-xz-zln(y)$; $(x,y,z)=(4,2,2)$

b$.a(x,y,z)=cosxsiny-z$; $(x,y,z)=(0,,1)$

c$.a(x,y)=x^2+y^2-16$; $(x,y)=(2,-2)$