Understanding Geostrophic & Ekman Flows: Thermal Wind & Margules in Atmosphere - Prof. Bay, Study notes of Oceanography

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Thermal Wind &

Margules Relations

Baylor Fox-Kemper ATOC Numbered Eqtns & Figures from Vallis

Thermal Wind

from Vallis or Fresh or Salty , buoyant (^) , not buoyant

The Momentum

Equations

Dv Dt

• f × v = −∇φ + bk −f v = − ∂φ ∂x horiz.-mom: Lo Rossby, Geostrophic f u = − ∂φ ∂y vert.-mom: hydrostatic ∂φ ∂z = b

Cross-Differentiated

Momentum Equations

Dv Dt

• f × v = −∇φ + bk horiz.-mom: Lo Rossby, Geostrophic vert.-mom: hydrostatic

2

∂z∂x

∂b

∂x

∂ 2 φ ∂z∂y = ∂b ∂y ∂f v ∂z = ∂ 2 φ ∂z∂x

∂f u

∂z

2

∂z∂y

from Vallis f × ∂v ∂z Thermal Wind =^ −∇b

Physical Ekman

Consider a Small Ro, O(1)=Ek balance, so Coriolis over a layer Friction over a layer Coriolis

0 = −f × v + F

Cushman-Roisin & Beckers Ekman Flow Roa = U f L ; Rot = 1 f T

Shallow Water Systems

Momentum Volume

Momentum Volume

Geostrophy in Shallow Water

Thermal Wind in Layered Systems: Margules Relation f × ∂v ∂z = −∇b f × ∆v ∆z = − ∆b ∆x

What happens in Layers? Barotropic & Baroclinic Cushman-Roisin & Beckers

What happens in Layers? Potential Vorticity Cushman-Roisin & Beckers