Sunday, March 27, 2011

Some Basics of Mountain Flow

It's been a week since my last blog post and that's mostly because I've been away on spring break this past week.  But I am back now, so I should start posting things regularly again.

A few people have asked me recently about the dynamics behind mountain flow--why we get this "rain shadow" effect among other things.  There are several excellent resources out there that explain this, but I'll do my best here to try and describe what's happening--and why flow over mountains becomes very, very complicated.

So, let's set the stage.  We'll assume we have a north-south running mountain chain (like the Rockies, the Sierras or the Cascades) and prevailing west to east flow (like usual).  The air in the lower parts of the atmosphere tends to have much more moisture than the air above it.  This is because the sources of water vapor are all near the ground--the oceans, lakes, evaporation from plants and so on.  The dotted line at the top of the figure below represents the stable tropopause layer.
(Click to enlarge this or any image)
So what happens when this air reaches the mountains?  Air in the lower levels will be forced up to rise over the mountains.  It can't go down because the ground is below (or, there's more air below that's also being forced up).  Therefore this lower-level air gets lifted over the mountains.
Remember that the low-level air is moister than the air above.  As this low-level air is forced upward, it cools.  Anyone who has ever climbed a mountain knows that it gets colder the further up you go.  As the air cools, it soon reaches its dewpoint temperature and the moisture in that air condenses into clouds and eventually rain.  This is why we see the heaviest rain on the windward slopes of mountains.  More on that later.

Anyhow, the air continues to move over the mountain range.  All that upward motion from below pushes on the air above it, forcing upward motion in the air above further downstream.  This causes a sort of  translation of upward motion up through the atmosphere--which will become important later.  In the meantime, eventually most of the moisture in the lower-level air is condensed out into precipitation as the air is forced up and over the mountains.  This air becomes significantly drier.
By the time the air reaches the far side (the leeward side--away from the direction of the prevailing wind) of the mountain, there's now a sort of "vacuum" below--as the terrain slopes down and away, the air rushes down the slopes of the mountain to fill in all that space.

In the meantime, that upward motion that has been translating up through the atmosphere has run into the tropopause.  Remember that the tropopause layer is very stable and tends to resist vertical motion.  We'll get back to what happens there later.  But now as the air rushes down the leeward side of the mountain, its pressure increases and it warms up.  What little moisture there was left in the air is now nowhere near saturation--as the air warms, the difference between the dewpoint of the air (the temperature at which it becomes saturated) and the actual temperature of the air will increase.  This makes the air feel very, very dry.

This dryness on the lee-ward side of the mountains is what gives rise to the "rain shadow" effect--all the precipitation tends to fall on the windward side of the mountains where lift is being forced by the terrain.  By the time the air has gotten over the mountains, not only has it lost much of its moisture, but the combination of downward motion and the air heating up (bringing the air further and further away from saturation) makes the leeward side of the mountains a poor place for any precipitation to develop at all.  This is starkly evident if we look at a map of annual precipitation in the western US:
Average annual precipitation in the western United States from 1961 to 1990.  From the Oregon State PRISM group.
The effects of major mountain ranges dictate where we see precipitation maxima.  I tried to annotate roughly where the crests of several mountain ranges are.  Note how particularly well they correspond with the edges between precipitation maxima (the blues and purples) and minima (the reds and yellows).  Particularly striking are the effect of the Cascades in the northwestern US and the Sierra Nevada in northern and eastern California.  To the west of these ranges there is heavy precipitation --80-100 inches of rainfall each year.  But just to the east of them, there are enhanced areas of dryness--only 5 or 6 inches of precipitation per year in some areas.  We see similar, but less intense separations over some of the mountain ranges further east like the Bitterroots in Montana and even the Wasatch in northern Utah.  There's still a maximum to the west of the mountain crest and a minimum to the east.

By the time we get to the high peaks of the Front Range of Colorado, though, there really isn't that much of a stark divide in precipitation--you can see hints of one, but its not nearly as obvious as it was further west. Why is that?  Each time air is forced over a mountain range, it loses more moisture to precipitation on the windward side of the mountains.  The most moist air is out over the Pacific Ocean--naturally.  Most of the moisture in that air is immediately lost when it hits the Coast Ranges, the Cascades or the Sierras in the far west.  The low-level air still retains some moisture, but not nearly as much by the time it crosses all those high mountains.  So by the time it is forced up and over ranges further east like the Bitterroots and the Wasatch, we still see some precipitation enhancement, but there was much less moisture to begin with.  And by the time the air finally reaches the Front Range of Colorado, it has very, very little moisture left in it--all the moisture has been "wrung out" by precipitation over multiple mountain ranges.  Thus we really don't see nearly as much of an enhancement in precipitation on the windward side of those far eastern ranges.  We do see a little, but not much.

If you're not interested in further technical details about forcings and responses in the atmosphere around mountains, then you can just stop here and enjoy seeing rain shadows across the US...

But now I want to go back to the schematics I've been drawing to look at a slightly more complicated question of what's causing what in this case.  The atmosphere is full of forcings and responses to forcings.  Mountains are particularly fun cases to look at.

First, what is the most direct effect the introduction of a mountain to our simple eastward-moving air?  If you put a mountain into the flow, the air is going to crash into the mountain on the windward side and there's going to be kind of a hole left in the flow (that "vacuum") on the leeward side.  If we look at this in terms of pressure, we would see higher pressure being caused by the air crashing into the mountain on the windward side, and lower pressure in that vacuum on the leeward, eastward side.  We'll consider this the "forcing" of the mountain.
Now the question becomes--what does the atmosphere do to respond to this forcing?  Well, it wants to evacuate air from the area of higher pressure and bring air in to fill the area of lower pressure.  So we see the response in terms of vertical motion--the atmosphere tries to remove air from the higher pressure are by drawing it up vertically (it can't pull down air because the side of the mountain is right there--and that's what's causing the higher pressure in the first place).  The atmosphere also tries to fill the low pressure area by pulling air down into it from above to increase the pressure.
This makes perfect sense because that's what we intuitively think of happening with the flow over a mountain--air generally rises on the windward side and falls on the leeward side--it goes up, over the mountain, then comes back down.  Furthermore, this agrees with how we think of precipitation--we almost always associate precipitation with rising motion, and that's exactly where we see it.

But it gets slightly more complicated in real life.  Usually the air is relatively stable while this is going on.  Up near the tropopause it's particularly stable.  Remember how that upward motion was being translated upward in the atmosphere?  Eventually that will reach the tropopause, usually just downstream of the mountains.  The very stable tropopause is not conducive to rising motion--it's very difficult to force air to move vertically beyond that height.  So with all this air moving upward toward the tropopause, the air will have to start diverging there because it can no longer move upward.

So now we see sustainable upward motion, supported by divergence aloft, directly over the area of downward motion trying to fill in the low pressure area on the downwind side of the mountains.

The net result is that some of the downward motion on the leeward side of the mountains is effectively "cancelled out" by the upward motion supported by divergence aloft at the tropopause.  This means that the atmosphere cannot "fill" the low pressure on the leeward side of the mountain--the downward motion that would do so is being somewhat cancelled out by the upward motion supported by the divergence aloft.  Thus, with this kind of flow pattern, there will always be an area of lower pressure on the leeward side of the mountains.  This is the "lee-side low" or the "downslope low" and its formation is referred to as lee-side cyclogenesis.

That's a pretty complicated system of forcings and responses.  There are two things going on--the direct response to the mountain being there and the indirect response of the atmosphere aloft to the presence of the mountain.  I'll try to summarize what happens in a list here:

  1. The presence of the mountain creates a partial vacuum or area of lower pressure on the downwind side of the mountain.
  2. Air moving over the mountain top rushes downward to fill that low pressure center.
  3. In the meantime, upward motion caused by the air rising on the windward side of the mountain translates upward and eastward, eventually reaching the very stable tropopause usually just down-wind of the mountain.
  4. When this upward-moving air reaches the stable tropopause, it can't move up anymore and spreads out instead, creating divergence aloft.
  5. This rising air and divergence aloft somewhat "cancels out" the sinking air trying to fill the low on the east side of the mountain.
  6. As a result, the low pressure center never can get filled in and a lee-side low persists.

I hope that description helps some of you understand the dynamics around mountains and why this flow is very complicated.  These lee-side lows actually form the basis for a lot of the storm systems we see across the central United States.  Some time soon I'll look at how we can see these lee-side lows in our observations.

As always, if you have any questions, email me at the address on the upper-right part of my blog page.  I always enjoy hearing from my readers...


  1. Hi Luke,

    Thanks for the lucid explanation.
    One question: how high does a mountain range have to be to develop this pattern?


  2. Good question, @diessoli.

    To answer that, I would turn to a concept known as the Froude Number. In essence, this is the ratio between the speed of the ambient environmental air and the speed at which internal gravity waves can move through the air. One of the ways that this number can be interpreted tells you whether air will tend to flow around a mountain or over a mountain. The greater the tendency for air to flow over the mountain, the better the chances of getting downstream mountain waves and seeing patterns like I described.

    One form of the Froude Number (there are many) is
    F = U/(N*H)
    Where U is the mean wind speed, N is the Brunt-Vaisala frequency (a measure of stability) and H is the height of the mountain. It turns out that for the Froude Number, 1 is a critical value. If F<<1, the air is very stable and vertical motions are dampened so the flow will have a difficult time rising above the mountain and generating those downstream waves. If F>>1, however, the flow can degenerate and cause hydraulic jumps and areas of strong turbulence. So we want to look for a space where the air is near resonance with the mountain, or when F is around 1.

    So if we set F=1 and solve for H, using a mean wind speed of, say, 15 m/s and an average Brunt-Vaisala frequency of .01 per second, we get H=1500 meters, or almost 5000 feet. This number will vary depending on the stability (which changes the Brunt-Vaisala Frequency) or the mean wind speed, but this gives a reasonable threshold value for mountain height to generate those conditions--we'd want mountains around 5000 feet in this case.

    Hope that helps. And thanks for the question!


    P.S. The AMS glossary has a definition of the Froude Number here:

  3. Hi Luke. I am trying to bridge what you write (which is clearly experience) with the Gas Law: PV =n RT
    Presure * Volume = a constant * Temperature
    As air is pushed up, I would expect pressure to *locally* increase causing the temperature to increase.
    I'd expect the shear size of the mountain and tropopause to act as a heat sink, lowering the air temperature of the rising air (like the coils of an air-conditioning lowering the temperature of the compressed gas), causing the condensation.
    I'd expect the the air to speed up and compress between the tropopause and the mountain tops like air passing through a nozzle. On the leeward side, i would expect the pressure and temperature to drop quickly, drying the air, as you say here.