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This page last updated on 04/17/2017.

Copyright 2001-2017 by Russ Meyer


Are you ready for a wild ride?  Great...then strap on your skates, we're goin' in!

Wind-Swept Observations

It was very windy today.  I was out walking in this gale for a while and began thinking about it.  A pressure gradient between centers of high and low pressure is what causes most wind.  Low pressure systems are sometimes referred to as "cyclones" and high pressure systems "anticyclones."  If the Earth didn't rotate, the wind would blow directly from the high to low pressure areas in a straight line.  However, since the Earth is actually rotating, Coriolis effects cause the wind to instead follow a curved path.  In the Northern hemisphere, the Coriolis effect causes winds to swirl counter-clockwise around low pressure centers and clockwise around high pressure areas.

The wind curves enough so that it blows almost parallel to the atmospheric pressure isobars.  You can use this knowledge to locate the direction of high and low pressure centers.  Stand with the wind at your back.  Raise your arms out to the left and right...so you look like a big letter "T."  Now rotate about 30 to your right (clockwise).  Your left arm is now pointing in the direction of the low pressure center and your right arm is pointing at the high pressure center.  You know that any bad weather associated with the low pressure area will be towards your left.  There should be good weather towards your right.  When I'm out flying, I often land at isolated little airports with no way to get weather information.  I've used this method to help me figure out where the bad weather is so I can avoid it on cross country flights.

A Thought

This is all very interesting, and I was mulling this over today as I trudged through the gale.  When I got home from work, I pulled out my meteorology book and looked all this stuff up, just to be sure I was remembering it correctly.  As I was reading about Coriolis effects and wind, a fabulous idea occurred to me.  The amount that the wind is influenced to curve from a straight line is related to the magnitude of the Coriolis effect.  The magnitude of the Coriolis effect is related to the rate of rotation of the Earth.  So, how much the wind curves is related to the rotation period of the Earth.  It should be possible to develop an equation to describe this.  The equation would relate four basic characteristics of any given cyclone or anticyclone:

  • Wind Speed (related to pressure gradient)
  • Cyclone or Anticyclone Diameter (balance between pressure gradient and Coriolis effect)
  • Latitude of the Cyclone or Anticyclone (impacts net Coriolis effect)
  • Planetary Rotation Rate (origin of Coriolis effect)

There are probably other factors that affect the equation, like surface roughness, atmospheric composition, average atmospheric temperature, etc.  However, I think the four factors above are probably the biggies.

Anyway, armed with this equation, you could calculate any one value given the other three.  For example, the rotation rate of a planet could be calculated given the wind speed, diameter, and latitude of any particular cyclone.  Someday I might fiddle with this a bit more to come up with an equation relating these four factors.  I could use satellite images of low pressure areas or hurricanes to test it.  (Low pressure areas have lots of clouds that make photometry easier.)  I suppose you could use surface pressure maps too.

Speculation

After thinking about this for a few minutes, another idea occurred to me.  Due to tidal effects of the moon, the length of an Earth day has been increasing since the Earth was formed 4.5 billion years ago.  Way back then, the length of an Earth day was about 14 hours.  This means the Coriolis effect was much greater in past epochs and would have affected the size of high and low pressure systems.  The world climate balance may have been significantly impacted by this.

I'm not sure, because I haven't calculated it, but I think a shorter day would have cause high and low pressure systems to be much larger in diameter.  There would have been a much stronger Coriolis effect trying to turn the wind parallel to the isobars.  I can see this having a several of possible effects:

  1. Persistent Atmospheric Pressure Differentials - This assumes the mean atmospheric wind speed is unaffected by the length of a day.  If that is true then movement of a parcel of air from high to low pressure areas would have been delayed.  The air parcel would have farther to travel because it was taking a more indirect route; a route more parallel to the isobars.  Delaying the flow of mass from high to low would have allowed atmospheric pressure differentials to be more persistent.
     
  2. Higher Mean Atmospheric Wind Speed - This assumes the flow of air mass from high to low pressure areas is unaffected by the length of a day.  If that is true then the mean atmospheric wind speed would have to increase.  The distance a parcel of air would have to travel from high to low would be greater because it would be taking a more indirect route; one more parallel to the isobars.  To maintain a constant mass flow, the mean wind speed would have to increase to allow the parcel of air to cover the larger distance in the same amount of time.  In effect, the increased in wind speed would enable overall mass flow from high to low to be held constant.
     
  3. Combination of #1 & #2 - Perhaps the effect was a little higher mean wind speed and longer lived pressure systems.
     
  4. Counterbalance of #1 & #2 - Perhaps the mean wind speed and duration of atmospheric pressure differentials counterbalance each other, allowing atmospheric characteristics to remain basically unaffected by the length of a day.

I favor scenario #2...just a hunch.  Here's why:  Wind is a symptom of a pressure differential in the atmosphere.  The pressure differential actually represents a heterogeneous mass distribution.  Because of the gravity gradient, the mass wants to flow so as to equalize it's distribution.  Like a tub of water, it all seeks the same level.  The wind is just a transport mechanism to equalize this disparate mass distribution.  It can be thought of as a hose connecting two tubs, one full of water and one empty.  As the water flows from one tub to the next, this hose conveys the mass.  When the length of a day is shortened, the Coriolis effect turns the wind more parallel to to isobars.  This lengthens the path that an air parcel has to travel from high to low pressure.  If the physical dimensions of the high and low pressure systems stay constant (e.g. constant outflow and inflow areas), then the aspect ratio of the wind transport "pipe" gets larger.  In other words, it looks like a skinnier pipe.  Because the mass transfer is, in effect, funneling through a smaller pipe, Bernoulli's law says the speed of the air parcel has to increase.  The mean wind speed increases.

 

In the time of the dinosaurs, the Earth was much warmer and wetter.  Much of the Earth was like a tropical rain forest.  Vegetation thrived and the atmospheric oxygen levels were correspondingly high.  Today oxygen makes up 20% of the air, but back then it was more like 30%.  That's what enabled there to be huge insects like dragonflies with 29 inch wingspans.  Perhaps these conditions made it possible for big dinosaurs to thrive...sort of turbo charging the biosphere.  Maybe this had something to do with the effect of a shorter period day (stronger Coriolis force) on the atmosphere.  I've never seen a climate study that ever took this factor into account when modeling ancient weather.  One other effect of a shorter day is a reduction in diurnal temperature variation.  When I calculate the length of a day about 80 million years ago, I get about 23 hours.  Not really very different from today, but I don't know how sensitive the atmosphere is to the length of the day.  All this could be simulated with a climate model or perhaps grossly estimated analytically.  It would take a lot of calculator work to slog through all the equations though.

Epilog:  The more I think about my conclusion that a shorter day would increase the mean atmospheric wind speed, the more I doubt it.  I think I am misapplying Bernoulli.  I'll have to mull it over some more.

Taking Another Look

OK, so I've been thinking about the above speculations for a couple of weeks, and I have another theory.  One which I believe is more useful.  Let's start all over again and think things through from a different perspective.

Let me just state up front what I think the real answer is:  The rotation rate of the Earth has been slowing down due to lunar tidal effects.  This has been going on since the Earth was first formed.  It means that in past epochs, the length of an Earth day was shorter than it is now.  Speeding up the rotation rate of the Earth increases the Coriolis effect.  This alters the nature of high and low pressure systems in two ways:

  1. It reduces the mean atmospheric wind speed.
  2. It allows pressure systems in the atmosphere to be more persistent.  In other words, high and low pressure centers took longer to dissapate.

I think this allowed weather systems to be more persistent and overall more mild.  That's a speculative leap of sorts.  Anyway, the following is an explanation of how I arrived at this conclusion.

It occurred to me that an atmospheric pressure differential is analogous to a charge separation in an electrical system.  Wind is like the flow of electricity.  The distance the wind has to flow is analogous to a resistor in series with the flow.  Rather than modeling the atmosphere like two tanks filled with fluid and connected by a pipe, it was even better to model it as a capacitor and resistor combination, like this:

                         

The heterogeneous distribution of mass in the atmospheric pressure system is modeled by the charge separation in the capacitor.  The atmospheric pressure differential is analogous to the voltage across the capacitor.  The amperage in the circuit is very similar to the mass transported by the wind.  The circuit resistance is similar to the distance the air parcel has to be moved by the wind.  This last point may be a bit obscure; I'm assuming there is a average amount of wind resistance per mile due to terrain roughness, etc. as the wind blows over the surface of the Earth.  To sum up:

  • Atmospheric pressure differential is analogous to voltage across the capacitor

  • Effective distance between pressure centers is analogous to resistance

  • Wind speed is analogous to amperage

As Coriolis effect increases, the wind blows more parallel to the isobars.  This increases the effective distance an air parcel has to travel to get from high to low.  In the electrical model, this effect is duplicated by increasing circuit resistance.  This causes a decrease in both circuit amperage and rate of capacitor charge decay.  If the analogy holds for the atmosphere, this would imply that the pressure differential would be more persistent and that mean atmospheric wind speed would decline.

This whole model seems much better to me than all that Bernolli stuff I was foolin' with earlier.  For example, correlating circuit resistance to the distance an air parcel has to move works well not only for Coriolis effects.  It also works for the case where high and low pressure centers are just physically further apart.  The further apart they are, the more resistance we get in our model.  The model amperage drops, predicting a lower wind speed in the atmosphere.  In the real world, separating high and low pressure systems causes the isobars to be further apart; there is smaller pressure gradient per mile.  This causes the wind speed to be less.  So, the model seems to pass the common sense test.

Quantification - Taking the Model Further

 If the electrical analogy is close, then there are a lot of mathematical equations governing the electric circuits that can be applied analogously to the atmospheric situation.  Let's look at a few of those.

Ohm's Law

Ohm's law relates amperage, voltage, and resistance in DC circuits and is given as E = I * R.  E is voltage, I is amperage, and R is resistance.  Any of these quantities can be computed if the other two are known.  Applying this to the atmosphere:

    Let:

            Ph = atmospheric pressure of the high
            Pl = atmospheric pressure of the low
            V = velocity of the wind
            D = distance between high and low pressure centers
            C = constant to take into account unit conversion and other slope calibration considerations such

    Then:

            Ph - Pl = C * V * D

Once calibrated, it should be possible to use the equation to predict maximum wind speed, distance between high and low pressure centers, and the difference in pressure between two pressure centers.  As presented above, the "D" term assumes a constant Coriolis effect.  The term could be extended to include the Coriolis effect on the net effective distance an air parcel must flow to travel from high to low.  Let's just call this new extended term De for "effective Distance."  It would actually be an equation relating planetary rotation rate to the effective distance traveled by a parcel of air (rotation rate -> Coriolis effect -> net effect on distance traveled).  Substituting De for D in the equation would allow atmospheric modeling over various epochs.  De could also be extended to take into account the friction of the wind with the surface of the Earth1, friction due to viscosity effects of the wind moving through the enveloping air mass, etc.  The equation as given above summarizes these effects in the coefficient C, but the model could be improved by explicitly accounting for these.

Time Constant

A time constant is an electrical concept that describes the rate at which a capacitor will charge or discharge through a resistance.  The capacitor discharge case is particularly applicable to the atmosphere model we're mulling over.  The discharge profile is given as E = E0 * e-t/RC.  E is the voltage across the capacitor at any given time, E0 is the initial starting voltage across the capacitor, e is the constant 2.714, t is the elapsed time, R is the resistance in ohms, and C is the capacitance in farads.

If the capacitor is charged, the voltage, E, will decrease logarithmically over time...like this:

Using this to model pressure systems implies that the rate at which the pressure differential changes can be computed.  The equation describing this could be modeled after the capacitor discharge equation.  It would have the following form:

    Let:

            Ph0 = initial atmospheric pressure of the high
            Pl0 = initial atmospheric pressure of the low
            Ph = atmospheric pressure of the high
            Pl = atmospheric pressure of the low
            D = distance between high and low pressure centers
            Mh0 = amount of air mass associated with the high pressure center
            Ml0 = amount of air mass associated with the low pressure center
            C = constant to take into account unit conversion and other slope calibration considerations such

    Then:

            Ph - Pl  = C *(Ph0 - Pl0) * e-t/D(Mh0-Ml0)

By analogy with the electrical version of the equation, the pressure differential between the two pressure centers over time could be predicted.  The duration of the effect the pressure systems have on the weather could be inferred.  I am not really sure how to compute Mh0 and Ml0.  I think it would be something like modeling the pressure centers as cylinders and calculating volume.  The diameter of the cylinders could be set using the mean pressure isobar between the high and low pressure systems as a delimiter.  The height of the cylinder would be related to the amount by which the pressure of the system was above or below some reference datum like the average atmospheric sea level pressure of 29.92 inches of mercury.

Going Even Further

The equations above could be applied to an ideal case where there was one high pressure center interacting with one low pressure center.  In real life, there are pressure centers of various types scattered all over the place.  Perhaps those could be modeled as series or parallel circuits, or perhaps as a network of systems analogous to many linear circuit components in a network.  Linear circuit analysis techniques could then be used to solve the model equations.

The surface roughness of the Earth plays a role in wind velocity calculations.  The problem is, the wind can flow over a wide variety of terrain with different friction characteristics.  Mountains have a very different effect on wind than ocean.  You could establish a model of the Earth's surface with different roughness indices to account for that; sort of a surface roughness topology.

Footnotes

1 - This is basically a surface roughness index modeled with a polynomial.  The polynomial would provide the "back pressure" or resistive force on the wind using the variables of wind velocity and surface characteristics.  So, De would be a summation of the distance between high and low pressure centers, the effect Coriolis effect has on the distance the wind travels, the contribution of surface friction to net wind resistance, and inter-airmass viscosity drag effects on the wind.