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CLIMATE/EARTH 401�Geophysical Fluid Dynamics�Dynamical Meteorology

Richard B. Rood

rbrood@umich.edu

https://clasp.engin.umich.edu/people/richard-b-rood/

https://openclimate.org/

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Definition of Geopotential

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Horizontal Momentum Equations

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Some basics of Earth’s atmosphere

Troposphere: depth ~ 1.0 x 10⁴ m

Troposphere

------------------ ~ 2

Mountain

Troposphere

------------------ ~ 1.6 x 10^-3

Earth radius

Scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

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Some basics of Earth’s atmosphere

Troposphere: depth ~ 900 mb

Do you know these units?

Pressure:

mb = millibars

hPa=hecto Pascals

Scale analysis tells us that most of the mass of the atmosphere is in the troposphere.

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Pressure Units

1000 mb
1000 hPa
Pa = Pascal = N/m² = Newton/m²
Newton = kg m/s²

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height

Longitude, our x

Temperature₁

Temperature₂

Pressure₁

Pressure₂

(P₂-P₁)/(x₂-x₁)

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Pressure altitude

Under virtually all conditions pressure (and density) decreases with height.
∂p/∂z < 0
That’s why it is a good vertical coordinate.
If ∂p/∂z = 0, then utility as a vertical coordinate falls apart.

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Horizontal Momentum Equations

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To use pressure as a vertical coordinate

What do we need.

Pressure gradient force in pressure coordinates
Way to express derivatives in pressure coordinates
Way to express vertical velocity in pressure coordinates

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Expressing pressure gradient force

We are going to use the hydrostatic relation.

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Integrate hydrostatic relation in altitude

Pressure at height z is force (weight) of air above height z.

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Concept of geopotential

Define a variable Φ such that the gradient of Φ is equal to g. This is called a potential function.

We have assumed here that Φ is a function of only z.

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Integrating with height

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What is geopotential?

Potential energy that a parcel would have if it was lifted from surface to the height z.

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Vertical Structure and Pressure as a Vertical Coordinate

Remember, the geopotential is defined as

From basic physics course

Gravitational Potential Energy:

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What is geopotential?

Potential energy that a parcel would have if it was lifted from surface to the height z.
It is analogous to the height of a pressure surface.

We seek to have an analogue for pressure on a height surface, which will be height on a pressure surface.

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Linking geopotential to pressure

Definition of specific volume

Ideal gas law

Hydrostatic relationship

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Remembering some calculus

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Define geopotential height (assumption of constant g = g₀)

Z₂-Z₁= Z_T≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition.