Example(Adapted from MMS (Optional) Cyclonic Air Motion Suppose that there is an area of low pressure in the Northern Hemisphere, so that the pressure force per unit mass on an air element is -Vp/p and is radially inwards. One would think that the air should rush in radially under this force to“ fill in the hole” eu Instead, the wind may be such that air moves in circular paths around the depression. The radial acceleration (ar)r'y?=-r02=-v2/r. The real force acting radially per unit mass is, as noted, -(1/p)dp/dr, nd, in addition, we would have to include the inertial forces due to Earth's rotation, namely, a Coriolis force x(o)r'y'2' per unit mass. The combined effect of gravity and Earth's centrifugal force acts in the direction of the local vertical. Therefore, the equation of motion in the radial direction is ug 1 dp +290 dr Here, Q u =n ey is the component of the angular velocity in the vertical direction. If we compare the magnitude of the acceleration term with that due to Coriolis effect, 29u ve see that for a given Qu, Coriolis' effect becomes important for large values of r/vg. Therefore, we consider two limits Large values of r/ve. This leads to the so called Geostrophic Winds. In this case, the acceleration term nall and the appl g equation becomes 0= I dp +29U Consider for instance ve=10 m/s and r in the range of 100 to 400 km. At a latitude of 42, we have QU A 4.9 x 10- rad/s, and dp/dr=-2p Quvg A 0.0012 N/m, or 1.2 mb per 100 km, a moderateExample (Adapted from MMS) (Optional) Cyclonic Air Motion Suppose that there is an area of low pressure in the Northern Hemisphere, so that the pressure force per unit mass on an air element is −∇p/ρ and is radially inwards. One would think that the air should rush in radially under this force to “fill in the hole”. Instead, the wind may be such that air moves in circular paths around the depression. The radial acceleration is then, (ar)x′y′z ′ = −r ˙θ 2 = −v 2 θ /r. The real force acting radially per unit mass is, as noted, −(1/ρ)dp/dr, and, in addition, we would have to include the inertial forces due to Earth’s rotation, namely, a Coriolis force −2Ω × (v)x′y′z ′ per unit mass. The combined effect of gravity and Earth’s centrifugal force acts in the direction of the local vertical. Therefore, the equation of motion in the radial direction is − v 2 θ r = − 1 ρ dp dr + 2ΩU vθ . Here, ΩU = Ω · eU is the component of the angular velocity in the vertical direction. If we compare the magnitude of the acceleration term with that due to Coriolis’ effect, 2ΩU vθ v 2 θ /r = 2ΩU r vθ , we see that for a given ΩU , Coriolis’ effect becomes important for large values of r/vθ. Therefore, we consider two limits: • Large values of r/vθ. This leads to the so called Geostrophic Winds. In this case, the acceleration term is small and the approximate governing equation becomes, 0 = − 1 ρ dp dr + 2ΩU vθ . Consider for instance vθ = 10 m/s and r in the range of 100 to 400 km. At a latitude of 42o , we have ΩU ≈ 4.9 × 10−5 rad/s, and dp/dr = −2ρΩU vθ ≈ 0.0012 N/m3 , or 1.2 mb per 100 km, a moderate 8