METEO 300
Fundamentals of Atmospheric Science

10.7 Natural coordinates are better horizontal coordinates.

Understanding the results of a balance of forces can often be easier if we choose a horizontal coordinate system that is aligned naturally with the air flow, and not just set up in Cartesian coordinates x and y or spherical coordinates λ and φ. We can choose one direction – let’s call it s – so that it is aligned with the streamline and is always parallel with the flow and a second direction to its left – let’s call it n for normal.

For the horizontal momentum equation without friction:

D V Dt =f k × V p Φ

where V = i u+ j v and p MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaFiaabaGaey4bIenacaGLxdcadaWgaaWcbaGaamiCaaqabaaaaa@3725@ is the horizontal gradient on a pressure surface, not a height surface.

Let’s look at each term in the equation and put it in natural coordinates.

The acceleration of the air parcel: D V Dt

There are two ways that V can change. It can change by changing its speed ( V=| V | ), DV Dt , which occurs in the streamline direction s, or by changing its direction, which occurs in the normal direction, n. Note that n points to the left of the velocity vector.

Remember your physics, which showed that the acceleration due to rotation is the velocity squared divided by the radius of curvature of the rotation. In this case, the rotation is the rotation of the air parcel as it moves horizontally over the Earth. R is the radius of curvature of the changing direction, and by convention, R > 0 if the curvature is in the counterclockwise direction, and R < 0 if the curvature is in the clockwise direction.

The Coriolis force:

f k × V MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiaadAgacaaMc8+aa8HaaeaacaWGRbaacaGLxdcacqGHxdaTdaWhcaqaaiaadAhaaiaawEniamaaBaaaleaacaWGibaabeaaaaa@3E90@

The n component equals fV MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTiaadAgacaaMc8UaamODamaaBaaaleaacaWGibaabeaaaaa@3821@ because when f > 0, the force is to the right of the motion (i.e., streamline) and is, thus, negative because positive n points to the left of the motion. When f < 0, the force would be to the left in the positive n direction.

The pressure gradient force (PGF):

p Φ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTmaaFiaabaGaey4bIenacaGLxdcadaWgaaWcbaGaamiCaaqabaGccqqHMoGraaa@3996@

The n component of p Φ MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTmaaFiaabaGaey4bIenacaGLxdcadaWgaaWcbaGaamiCaaqabaGccqqHMoGraaa@3996@ is just Φ n MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabgkHiTmaalaaabaGaeyOaIyRaeuOPdyeabaGaeyOaIyRaamOBaaaaaaa@3901@ .

Putting these pieces all together results in the equation for the n-component of the horizontal momentum equation:

V 2 R =f V Φ n MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaamODamaaBaaaleaacaWGibaabeaakmaaCaaaleqabaGaaGOmaaaaaOqaaiaadkfaaaGaeyypa0JaeyOeI0IaamOzaiaadAhadaWgaaWcbaGaamisaaqabaGccqGHsisldaWcaaqaaiabgkGi2kabfA6agbqaaiabgkGi2kaad6gaaaaaaa@41B4@

[10.34]

This equation is called the Gradient Wind Equation.

Let’s rearrange this equation slightly by moving the centripetal force over to the right hand side, where it represents a centrifugal force. Then, by considering which of the terms are the most important, we can apply this equation to different situations:

V 2 R f V Φ n =0gradientbalance fV Φ n =0geostrophicbalance V 2 R Φ n =0cyclostrophicbalance V 2 R f V =0inertialbalance MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@D214@

[10.35]

A measure of the importance of the centrifugal force versus the Coriolis force is the Rossby number. If the force is solely centrifugal, then the Rossby number is:

R o = acceleration Coriolis = V fR MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkfadaWgaaWcbaGaam4BaaqabaGccqGH9aqpdaWcaaqaaiaadggacaWGJbGaam4yaiaadwgacaWGSbGaamyzaiaadkhacaWGHbGaamiDaiaadMgacaWGVbGaamOBaaqaaiaadoeacaWGVbGaamOCaiaadMgacaWGVbGaamiBaiaadMgacaWGZbaaaiabg2da9maalaaabaGaamODamaaBaaaleaacaWGibaabeaaaOqaaiaadAgacaWGsbaaaaaa@4D4A@

If Ro << 1, then the centripetal force is much smaller than the Coriolis force and Coriolis force balances the PGF (Geostrophic balance).

If Ro >> 1, then Coriolis force is relatively small and the centrifugal force and PGF are in balance (Cyclostrophic balance).

Quiz 10-2: Coordinates and scales.

  1. Find Practice Quiz 10-2 in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 10-2. You will be allowed to take this quiz only once. Good luck!