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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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Vertical Velocity Estimates

  • Important for
    • Clouds and precipitation
    • Severe weather
    • Cyclogenesis (spin up of vorticity through stretching)
  • Computed from governing equations

​

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Scaled equations in pressure coordinates �(The quasi-geostrophic (QG) equations)

momentum equation

continuity equation

thermodynamic�equation

geostrophic wind

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Methods for Estimating Vertical Velocity

  1. Kinematic method (continuity equation)
  2. Adiabatic method (thermodynamic equation)
  3. Diabatic method (thermodynamic equation)
  4. Omega equation (later in quasi-geostrophic theory)

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1. Kinematic Method:�Link between ω and the ageostrophic wind

Continuity equation (pressure coordinates) and non-divergence of geostrophic wind lead to

which can be�rewritten as:

​

​

�and solved for:

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Vertical Velocity: Problems

1. Kinematic method: �divergence 🡪 vertical motion

    • Links the horizontal and vertical motions. Since geostrophy is such a strong balance, the vertical motion is linked to the divergence of the ageostrophic wind, which is small difference of small numbers that are difficult to calculate.
    • Therefore: small errors in evaluating the winds <u> and <v> lead to large errors in ω.
    • The kinematic method is inaccurate.

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2. Adiabatic Method�Link between ω and temperature advection

Thermodynamic equation:

Assume the diabatic heating term J is small (J=0), and there is no local time change in temperature

Horizontal temperature �advection term

Stability parameter

warm air advection: ω < 0, w ≈ -ω/ρg > 0 (ascending air)

cold air advection: ω > 0, w ≈ -ω/ρg < 0 (descending air)

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2. Adiabatic Method�Link between ω and temperature advection

Based on temperature advection, which is dominated by the geostrophic wind, which is large. Hence this is a reasonable way to estimate local vertical velocity when advection is strong.

θ - Δθ

θ + Δθ

θ

WARM

COLD

adiabatic = no change in θ

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Vertical Velocity: Problems

2. Adiabatic method:

�temperature advection 🡪 vertical motion

    • Assumes steady state (no movement of weather systems)
    • Assumes no diabatic heating (no clouds or precipitation)
    • What about divergence/convergence?
    • The adiabatic assumption is limited.

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3. Diabatic Method�Link between ω and heating/cooling

Start from thermodynamic equation in p-coordinates:

Diabatic term

If you take an average over space and time, then the advection and time derivatives tend to cancel out.

Radiation�Condensation�Evaporation�Melting�Freezing

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mean meridional circulation

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Vertical Velocity: Problems

3. Diabatic method:

�Heating/cooling 🡪 vertical motion

    • Assumes definition of some average atmosphere
    • Assumes vertical motion only due to diabatic heating
    • What about divergence/convergence?
    • The diabatic assumption is limited.

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Vertical velocity summary

  • Hadley cell
    • Intertropical convergence zone
    • Deserts
  • Mean mass circulation
  • Mid-latitude cyclones
  • Convective systems

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End: Vertical Velocity Estimates

Richard B. Rood

rbrood@umich.edu

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

https://openclimate.org/