Midterm: UNIT TWO

Question 1

What is the expression for Rossby number?

Also express it in terms of time-scales

Answer 1

Ro = acc/coriolis. = U/fL
Ro_T = 1/fT_ph, where T_ph is timescale associated with phenomenon at hand
= Tin/T_ph, where T_in is the inertial timescale (~3hr)

Question 2

What is the assumption for geostrophic balance?

Answer 2

Ro &laquo_space;1
Balance between coriolis force and pressure gradient force

Question 3

Write geostrophic wind equation in vector format.
What is it?

Answer 3

u_g = 1/(f_0 rho) k x ∇h p

Theoretical wind that satisfies the above relation, approximates real wind to about 85%-90%

-ccw around a pressure minimum
-cw around a pressure maximum

Question 4

What is the divergence of geostrophic wind?

Answer 4

Geostrophic flow is non-divergent

Question 5

Link Helmotz decomposition with geostrophic wind

Answer 5

If non-divergent flow –> entirely described by streamfunction

Geostrophic streamfunction = p/(f_0 rho) such that the geostrophic flow is aligned along streamlines of constant 𝜳𝒈

Question 6

What is the vertical component of geostrophic vorticity?

Answer 6

ζ𝑔 = k (∇ × u_g) = ∇ℎ^2 Ψ𝑔

Question 7

Why should we use pressure coordinate system instead of height z?

Answer 7

-Radiosondes motivation
-monotonically varies in the vertical direction
-direct relationship b/w pressure at certain level and the mass of atm column of unit surface area above that elvel
-geostrophic and continuity equation have advantageous expressions in p-coord.

Question 8

What is the material derivative in p-coords?

What is the vertical velocity in pressure coordinates analogy?

Answer 8

• 𝐷𝑡 = 𝜕/𝜕𝑡 + 𝒖𝒑 ∙ ∇𝑝 + 𝜔 𝜕/𝜕𝑝
  where up and ∇𝑝 are taken on a pressure surface
• 𝜔 = 𝐷𝑝/𝐷𝑡

Question 9

Derive height to pressure coordinate

Answer 9

Make the triangle
what is (𝜕𝑝/𝜕𝑥)_z? (𝜕𝑧/𝜕𝑥 )_p? rewrite in terms of hydrostatic equation

–> 1/𝜌 ∇𝑧𝑝 = ∇𝑝Φ, ∇𝑝Φ are geopotential height contours that are plotted on constant pressure surfaces in upper troposphere

Question 10

Derive Hypsometric equation

Answer 10

Assume hydrostatic balance

integrate both sides for 𝑑𝑧 = − 𝑅𝑇/𝑔 𝑑𝑝/𝑝
–> tilt of pressure surfaces from Warm to Cold regions

Question 11

What is momentum eq in pressure coordinates?

hydrostatic equation?

Answer 11

• 𝜕𝒖𝒑/𝜕𝑡 + 𝒖𝒑 ∙ ∇ℎ𝑝 𝒖𝒑 + 𝜔 𝜕𝒖𝒑/𝜕𝑝 + 𝒇 × 𝒖𝒑 = −∇𝑝Φ
• 𝜕Φ/𝜕𝑝 = − 𝑅𝑇/𝑝

Question 12

Derive continuity equation in p-coordinates

Answer 12

Rewrite 1/𝜌𝛿𝑉 𝐷(𝜌𝛿𝑉)/Dt in p-coordinates
–> ∇𝑝 ∙ 𝒖𝒑= 0

-no assumption of incompressibility
- calculation of 𝜕𝜔/𝜕𝑝

Question 13

Write geostrophic flow in terms of geopotential heights

Answer 13

𝒇 × 𝒖𝒑 = −∇𝑝Φ

-Geostrophic flow streams along z contours on a pressure surface
“Streamlines of g. height z”

Question 14

Derive thermal wind balance

Answer 14

From geostrophy in p-coordinates
hydrostatic in p-coordinates

Take 𝜕 / 𝜕𝑝

you get

𝑓 × 𝜕𝑢𝑝/ 𝜕𝑝 = 𝑅/𝑝 ∇𝑝(𝑇)

A horizontal temperature gradient on isobaric surface is accompanied by a vertical shear of the geostrophic wind (i.e. by thermal wind)

Question 15

What are temperature contours?

Answer 15

They are streamlines for the thermal wind in a similar way that z contours are streamlines for geostrophic wind

Question 16

Departures from geostrophy
- express u in terms of geostrophy and ageostrophy
- what is Uag scaling?

Answer 16

𝜕𝒖/𝜕𝑡 + 𝒖 ∙ ∇ 𝒖 + 𝑤 𝜕𝒖/𝜕𝑧 + 𝑓 × 𝒖𝒂𝒈 = 0

–>scale such that fUag ~ U^2/L –> Uag ~ U^2/Lf = Ro U

So Uag is by a Rossby number smaller than the geostrophic wind

Question 17

For incompressible flow, derive what is 𝜕𝑤/𝜕𝑧 ?

scale W in terms of Ro

Answer 17

Use continuity equation, but geostrophic wind is non-divergent, you get
𝜕𝑤/𝜕𝑧 = −∇ℎ𝒖𝒂𝒈

W ~ Ro UH/L

so w is by an order Rossby number smaller than what would be expected from continuity eq. in the initial continuity form.