Why is the troposphere 8km higher at the equator than the poles?

Posted: September 16, 2012 by Rog Tallbloke in Astrophysics, atmosphere, Cycles, data, general circulation, Geology, Gravity, Measurement, Tides

This is an important question, because it is thought to be due to temperature, convection and radiative physics. But what about other possible factors? I’ve started gathering a list and some notes for discussion on the excellent and still very active Clive Best lapse rate thread. Regulars know that when a post title contain a question mark, there are going to be demands for help… TB

Dr Tim Ball has a good recent article here, which contains some useful diagrams, such as this one:

 

Tim says:

Tropopause height at the Poles varies between 7 km in winter and 10 km in summer, at the Equator the range is 17 to 18 km. The difference in seasonal range is because of the difference in seasonal temperature range. How do you build even those simple dynamics into a computer model?

Answer: the best way to make a start is to quantify the various effects which contribute to the height difference in the polar and equatorial tropopause, using the clues offered by seasonal variation as a guide.

So, what else apart from temperature, convection and radiation might affect the height of the tropopause?

Here’s my probably list, I’ll update it with suggestions made in comments.

Earth’s Rotation:
The Earth is 43.5km bigger in diameter at the equator than it is at the poles. Wikipedia says this is due to rotation; i.e. the centrifugal effect of the spinning mass.

“Without any idea of the Earth’s interior, we can state a “constant density” of 5.515 g/cm³ and, according to theoretical arguments (seeLeonhard EulerAlbert Wangerin, etc.), such a body rotating like the Earth would have a flattening of 1:230.

In fact the measured flattening is 1:298.25, which is more similar to a sphere and a strong argument that the Earth’s core is very compact.”

Of course, for a centrifugal force to be effective, there must be a coupling between the atmosphere and Earth. There clearly is some, because at the surface, the atmosphere goes round at the same speed the Earth does, give or take local variation. If it didn’t, our brollys would be blown inside out by 1000mph winds. However, there is shear as we go up from the surface, and jet streams, geostrophic winds, meridional circulation etc. So how do we quantify the effect?

Gravity:
If the Earth was on it’s own in space, then notwithstanding other effects the atmosphere would be pulled towards sphericity. However, Earth lies in a planetary plane orbiting the Sun, with a close by moon 1/6th its mass. About.com has this to say:

“The gravitational pull of the moon and the sun creates tides on the earth. While tides are most commonly associated with oceans and large bodies of water, gravity creates tides in the atmosphere and even the lithosphere (the surface of the earth). The atmospheric tidal bulge extends far into space but the tidal bulge of the lithosphere is limited to approximately 12 inches (30 cm) twice a day.

The moon, which is approximately 240,000 miles (386,240 km) from the earth, exerts a greater influence on the tides then does the sun, which sits 93 million miles (150 million km) from the earth. The strength of the sun’s gravity is 179 times that of the moon’s but the moon is responsible for 56% of the earth’s tidal energy while the sun claims responsibility for a mere 44% (due to the moon’s proximity but the sun’s much larger size).”

But we would expect the atmosphere to relax back towards sphericity (notwithstanding other factors) where it is not in line with these heavenly bodies. So how fast does it relax? How far is “far out into space”?

Entrainment:
Clearly the Earth’s topography has a hand in keeping the near surface atmosphere moving round with the surface, indeed there are clear inverse relationships between changes in atmospheric angular momentum (AAM), zonal Atmospheric Circulation Index ACI and the Earth’s length of day (LOD). So large amounts of coupled energy are evidence of the coupling between the atmosphere and solid Earth. What effect does this have on the difference in height of the troposphere between equator and poles? The north pole is pretty flat, the northern hemisphere is pretty mountainous and forested in places. The southern hemisphere is mostly ocean, but big waves will also couple the air and sea. Is there a difference in tropopause profile between the hemispheres which offers any clues here? Or would its signal be lost in profile differences due to larger effects such as general circulation?

Over to you.

Comments
  1. omnologos says:

    How about the temperature? It cannot be the same, as 7km=40C from lapse rate alone.

    The tropopause temp must be considerably lower at the Equator than at the Poles.

  2. tallbloke says:

    Sure, temperature obviously plays a big role, as I said in my intro, and as the seasonal variations indicate. But are the other factors I’ve listed negligible, or wrongly ignored?

  3. tallbloke says:

    Hey Joe Lalonde! You can join this thread! :)

  4. Michael Hart says:

    TB, can you give us a definition of how the troposphere is defined and measured [for the purposes of this article]?

  5. tallbloke says:

    Good question Michael. I would say, ‘Where the temperature stop falling with altitude and starts rising.

  6. tallbloke says:

    Just a note of caution for those having a think about the seasonal differences noted by Tim Ball.

    At the equator, the difference is primarily due to the elliptical nature of Earth’s orbit. The TOA insolation is ~37 W/m^2 greater at perihelion (closest approach to the Sun) than at aphelion.

    At the poles the primary effect is due to obliquity, the ~23 tilt of the Earth WRT to plane of it’s orbit around the Sun (ecliptic). This causes the area in the polar circles to be in darkness in winter and sunshine in summer.

    This makes for an interesting insolation map, which I’ll find and put in the post.

  7. tallbloke says:

    The density of a substance (mainly gases) depends on temperature and pressure. Gases are usually compared at a standard temperature and standard pressure. These are the freezing point (0 °C) and normal air pressure at sea level (760 torr), respectively.

    The density of dry air at sea level is 1.2929 kg/m3 or about 1/800th the density of water. But as altitude increases, the density drops dramatically. This is because the density of air is proportional to the pressure and inversely proportional to temperature. And the higher you go into the atmosphere, the lower the pressure gets. Pressure is approximately halved for each additional increase of 5.6 km in altitude. To determine the density of dry air at a given altitude we could use the relation

    D = D0 × (T0 / T) × (P / P0)

    Where D0 is the known density at absolute temperature T0 and pressure P0 and D, the unknown density at absolute temperature T and pressure P.

    Just as there is a density of dry air, there is also the density of moist air, or air that contains moisture (humidity). To obtain this density you can use the relation

    D × (273.15 / T) × [(B–0.3783 e)/760]

    Where …

    D is the density of dry air at sea level,
    T is the absolute temperature in kelvin,
    B is the barometric pressure in torr, and
    e is the vapor pressure of the moisture in the air in torr.

    I think we can take from this the conclusion that the amplitude of the tidal effect in the upper atmosphere is more strongly affected by the Luni-solar gravity than the lower atmosphere.

  8. tallbloke says:

    Wikipedia on tides:
    “The theoretical amplitude of oceanic tides caused by the moon is about 54 centimetres (21 in) at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were rotating in step with the moon’s orbit. The sun similarly causes tides, of which the theoretical amplitude is about 25 centimetres (9.8 in) (46% of that of the moon) with a cycle time of 12 hours. At spring tide the two effects add to each other to a theoretical level of 79 centimetres (31 in), while at neap tide the theoretical level is reduced to 29 centimetres (11 in). Since the orbits of the Earth about the sun, and the moon about the Earth, are elliptical, tidal amplitudes change somewhat as a result of the varying Earth–sun and Earth–moon distances. This causes a variation in the tidal force and theoretical amplitude of about ±18% for the moon and ±5% for the sun. If both the sun and moon were at their closest positions and aligned at new moon, the theoretical amplitude would reach 93 centimetres (37 in).”

    Given that at sea level water is ~800 times denser than air, and ~1600 times denser than air at 5.6km, and ~3200 times denser than air near the tropopause, it would seem that the Sun and Moon together might cause a not inconsiderable portion of the difference in the polar and equatorial tropospheric height. They must be responsible for significant tidal mixing too. Where is the research on this?

  9. Stephen Wilde says:

    You need to also take into account longer term multidecadal and centennial influences such as those from ocean cycles and solar variations.

    They go beyond normal seasonal changes to produce longer term climate zone shifting which is related to the slope of the tropopause height between equator and poles.

    A warm ocean cycle such as a positive PDO will push the tropopause upwards. A cold ocean cycle will allow it to drop.

    Solar variations appear to alter the temperature of the stratosphere from above which can pull the tropopause up if the stratosphere cools (more positive AO and AAO) or push it down if the stratosphere warms (more negative AO and AAO).

    The solar effect seems to be more pronounced at the poles and so skews the slope of tropopause height between equator and poles to cause climate zone shifting which alters global albedo via cloudiness changes.

    I’d guess that tidal effects will be in there somewhere but small as compared to the solar effect on the polar vortices and on much shorter timescales.

    You might be able to use tidal forces to help with weather prediction but for centennial scale climate zone shifting it has to be mainly solar effects.

    The solar effects could link to changes in the centre of gravity of the solar system though.

  10. tallbloke says:

    Hopefully, Richard Holle will drop by with his latest findings on lunar influence. One thing worth noting is that due to the inverse square nature of gravitation, the changing distance of the moon from the Earth causes up to a 50% change in the G-force exerted. So along with the variation caused by the monthly lunar orbit and the bi-decadal declination cycle, we have a nice complex pot stirrer for the weather systems. Maybe a way to test for its influence would be to combine the lunar cycles with the annual ~37W/m^2 feature of the solar relation and see what signal drops out of a local area known to exhibit the 18.6 year drought cycle.

  11. Ian W says:

    The tropopause is where the tropo (movement) pauses (stops). In other words it is where convection will cease usually because it is the cross over point of dry and wet adiabatic lapse rates. However, the convection in the tropics and in the Inter Tropical Convergence Zone is far stronger than some people realize with updrafts of over 100Kts carrying liquid water seven miles or more upwards in the atmosphere where it freezes adding latent heat to the surrounding air increasing/maintaining the updrafts. (Note – aircraft are warned to stay 20 miles away from severe convective weather as hail can be thrown 10 – 15 miles from the top of convective storms.) So there is a significant kinetic effect from updrafts rather like watching the surface of water boiling in a pan. If the heat is turned down then the convection reduces in strength and the tropopause reduces in altitude.

    The surface of the troposphere (the tropopause) is not flat and waves can travel along it and up into it. Kelvin waves in the tropics cause undulations in the Tropopause:

    Abstract
    The relationship between local convection, vertically propagating Kelvin waves, and tropical tropopause height variability is examined. This study utilizes both simulations of a global primitive-equation model and global observational datasets. Regression analysis with the data shows that convection over the western tropical Pacific is followed by warming in the upper troposphere (UT) and cooling in lower stratosphere (LS) over most longitudes, which results in a lifting of the tropical tropopause. The model results reveal that these UT–LS temperature anomalies are closely associated with vertically propagating Kelvin waves, indicating that these Kelvin waves drive tropical tropopause undulations at intraseasonal time scales.

    The model simulations further show that regardless of the longitudinal position of the imposed heating, the UT–LS Kelvin wave reaches its maximum amplitude over the western Pacific. This result, together with an analysis based on wave action conservation, is used to contend that the Kelvin wave amplification over the western Pacific should be attributed to the zonal variation of background zonal wind field, rather than to the proximity of the heating. The wave action conservation law is also used to offer an explanation as to why the vertically propagating Kelvin waves play the central role in driving tropical tropopause height undulations.

    The zonal and vertical modulation of the Kelvin waves by the background flow may help explain the origin of the very cold air over the western tropical Pacific, which is known to cause freeze-drying of tropospheric air en route to the stratosphere.

    http://journals.ametsoc.org/doi/abs/10.1175/2007JAS2466.1

    Rossby waves in the northern hemisphere are caused due to orographic effects propagate up to the tropopause. These Rossby waves can ‘break’ leading to ‘Sudden Stratospheric Warming’ events which can temporarily halt or even reverse the polar vortex.

    http://www.atmos.umd.edu/~seminar/past_files/misc/Norton1994.pdf and http://en.wikipedia.org/wiki/Sudden_stratospheric_warming)

    There is a tendency among the more mathematically minded to see the Tropopause as a flat surface between two distinct slabs of atmosphere the troposphere and stratosphere. However, it is probably closer to think of them more in terms of the behaviour of a lava lamp ;-)

    However to return to the boiling water analogy – imagine a large pan with the heat being applied at the centre only. As it reaches the boil the water convects up lifting the surface in the centre over the heat source. This is precisely what is seen in the atmosphere; the heat source being the hot tropical sea surface and land surfaces leading to convection and ‘boiling up’ of severe convective weather.

  12. tallbloke says:

    Great post Ian W.
    I still don’t think the temperature, convective inertia and ‘radiative sinking’ account for all the height difference though. For one thing, more spaceward radiation takes place above the equator than the poles, so this should diminish the height difference if anything.

  13. Clive Best says:

    The tropics also mainly follow a moist lapse rate. This is governed by release of latent heat as water vapour evaporated from the surface condenses out releasing heat to surrounding air. Then there are regular thunderstorms triggered at the end of the day as air becomes unstable. Saturated moist lapse rate is 5 deg/km just half the dry adiabatic lapse rate. I suspect then that the temperature at the tropopause is the same at the tropics as it is in a dry atmosphere (poles) – but I have direct no evidence at hand. If true then the tropopause in the tropics would be naturally 50-100% higher than it is at the poles.

    Regarding effects of the moon – For sure the moon does have an effect on climate. The most important effect is to stabilise the Earth’s axis of rotation, giving stable seasons. Tidal forces fall off as 1/R^3 which explains why why the moon has a larger tidal effect than the sun. The inclination of the moon also varies with an 18.6 year cycle, so the position and strengths of tidal bulge vary with this and the earth’s orbit. Some studies claim to have observed an 18.6 year cycle in droughts for large continents (China and US). Atmospheric tides are dominated by solar insolation effects rather than gravitational tides. I think that gravitational tides though at the poles especially in winter could be significant. Tides work more by dragging fluids tangentially perpendicular to gravity rather than directly against gravity. So I don’t think gravitational tides as such really effect much the height of the atmosphere.

  14. tallbloke says:

    Clive: I think you need to follow your statement through to the logical conclusion:

    Tides work more by dragging fluids tangentially perpendicular to gravity …

    These horizontal components in concert with planetary rotation cause the bunching which produces the vertical component of the tides, which in the case of the moon acting on a medium more than a thousand times less dense than water, will be of considerable height.

  15. olsonjs444 says:

    Good posts, all. There is at least one other effect which must be evaluated from empirical evidence, and either investigated further or proven to be inconsequential.

    This effect is the change in pressure due to dynamics in the ionosphere, which are a complex interaction between electromagnetic forces and solar phenomena. The electromagnetic forces are not trivial: solar flares and coronal mass ejections create immediate, observable perturbations, but charge accumulation over long periods of time can also generate instantaneous discharges or “storms” in the proton belt. These give us our aurora.

    It is already established that periods of solar minima are correlated with the proton belt growing thicker; however, it is unclear whether this increases or reduces the pressure on the top of the atmosphere. Electrodynamics direct protons in helical paths along the magnetic field lines. The protons collide with molecules in the upper atmosphere and “bounce” back into the ionosphere. This is why they are referred to as “trapped protons” by those of us who must evaluate radiation effects on satellite electronics.

    The thicker belt could be due to more protons or due to reduced pressure (fewer protons). The Radiation Belt Storm Probes were successfully inserted into orbits through the proton & electron belts a week or two ago, and will measure these effects more accurately over the next two years.

    It has been assumed the growth in the Van Allen belts occurs due to a reduction in pressure from the solar wind, which is less energetic during solar minima. Similarly, periods of solar maxima are associated with reduced proton belt thickness, allegedly due to greater solar wind pressure. What is known is low earth orbiting satellites and the space station require more orbit corrections during periods of solar minima. This is because the atmosphere expands into the orbits of these low earth satellites and creates greater drag. It is hard to conceive how this expansion could occur without a concurrent reduction in pressure – however slight.

    Solar wind energy has already been measured with sufficient accuracy to determine it cannot be a significant factor impacting the height of the troposphere, but I’m not aware of any scientific evaluation of the indirect effects due to earth’s magnetic field dynamics or due to the thickness of the proton belt (trapped charge). My gut says the answer will come back as “it’s not a factor”, but I just don’t know.

  16. tallbloke says:

    Olson: Interesting and informative, thanks.
    Adolfo, good catch, I’d lost that one.
    Also, there’s the NASA story about the thermosphere (?) shrinking 30% after 2003

  17. Clive Best says:

    @TB
    I think the tidal bulge in the high upper atmosphere > 50 km is very significant. Tidal winds above 80 km are large. But in the troposphere lunar tidal effects are swamped by solar insolation diurnal thermal “tides”. However their effects are still there and may be more subtle…
    eHow: “Satellite measurements of the temperature of the atmosphere shows that the poles are 0.55 degrees Celsius (0.99 degrees Fahrenheit) warmer during a full moon than during a new moon.”

    We often hear the remark that weather systems are chaotic and a hurricane can be triggered by the flapping of butterflies’ wings. So the lunar tides may be one trigger for stormy weather, especially as tides are stronger nearer the poles and vary inclination every 18.6 years.

  18. Michael Hart says:

    Isn’t the tropopause the point where heating from above effectively matches heating from below [using the term loosely]? I see no mention of Ozone and attendant factors such as spectral variation/scattering etc. At high albedo I would guess that increased UV will be coming from the surface. Or are these things already included implicitly?

  19. Dare I suggest that the low level of the tropopause at the poles during the winter follows from the near zero concentration of water vapor which means that the dry adiabat rules?

    Conversely, at the equator there is a relatively huge concentration of water vapor so the moist adiabat rules.

    I think you all know that the dry adiabatic lapse rate is higher than the moist one.

  20. Stephen Wilde says:

    The entire atmosphere is less deep at the poles as compared to at the equator.

    How about more insolation at the equator adding more energy to the atmosphere so that it expands to a greater height ?

    We saw previously that insolation is what prevents an atmosphere from congealing on the surface and the more solar energy the higher it can go.

    Of course, the horizontal transfer of energy by air and oceans reduces the differential but it remains large.

  21. tallbloke says:

    In polar summer insolation is greater than it is at the equator, because it’s happening 24 hours a day.
    Here’s the map:

    ~ 600 W/m^2 obliquity caused variance in insolation makes a difference of 3km to the polar troposphere over the year.
    The Winter/summer air temperature varies considerably causing a 30% variance in tropopause height

    ~37 W/m^2 perihelion/aphelion variance makes a difference of 1km to the equatorial troposphere over the year.
    The perihelion/aphelion air temperature varies not a lot causing a 6% variance in tropopause height

    It would be interesting to know the difference between tropopause height variation in the arctic and antarctic, along with the winter/summer air temperatures. That might tell us more about the degree to which tropopause height is determined by temperature.

    We’d probably need to know more about cloud cover variation over the year at the two polar locations too though.

    Simple it aint.

  22. suricat says:

    I can see that none of you guys are sailors, or even coarse fishermen!

    tallbloke says: September 16, 2012 at 1:31 pm

    “Wikipedia on tides:
    “The theoretical amplitude of oceanic tides caused by the moon is about 54 centimetres (21 in) at the highest point, which corresponds to the amplitude that would be reached if the ocean possessed a uniform depth, there were no landmasses, and the Earth were rotating in step with the moon’s orbit……”

    Whatever ‘Wikapedia’ says about the theory, ocean/sea tidal span is measured in metres and not centimetres! Are you sure your source doesn’t relate to ‘crustal movement’ TB? Here’s a link to a UK authority, and note that ‘tidal movement’ in Earth’s oceans and seas is ‘sinusoidal':
    http://www.pol.ac.uk/ntslf/networks.html

    Best regards, Ray.

  23. gallopingcamel says:

    suricat, September 17, 2012 at 11:55 pm

    Having operated a fish farm on the bay of Fundy I know what you mean. At Eastport, Maine the tidal amplitude averages about 6 meters peak to peak. Enjoy this little video:
    http://www.liveleak.com/view?i=7f9_1271371289

    The average theoretical equatorial ocean tidal amplitude is 0.54 meters but there is a wide variation between spring tides (~0.79 meters) and neap tides (~0.39 meters).

    These apparently conflicting numbers can be reconciled but it involves more mathematics than I can handle. Fortunately Texas A&M has done the math so the rest of us won’t have to:
    http://oceanworld.tamu.edu/resources/ocng_textbook/chapter17/chapter17_05.htm

  24. oldbrew says:

    This may be of interest.

    http://www-das.uwyo.edu/~geerts/cwx/notes/chap01/tropo.html

    Quote: “colder regions have a lower tropopause, obviously because convective overturning is limited there, due to the negative radiation balance at the surface”

  25. tallbloke says:

    GC: The atmophere doesn’t have continents in it, so the theoretical mid ocean- no continents scenario seemed like the reasonable startng point for calculating ‘airtide’.

    Oldbrew, I usually disregard science that includes the word ‘obviously’. ;)

  26. Ian W says:

    olsonjs444 says:
    September 16, 2012 at 7:00 pm

    It has been assumed the growth in the Van Allen belts occurs due to a reduction in pressure from the solar wind, which is less energetic during solar minima. Similarly, periods of solar maxima are associated with reduced proton belt thickness, allegedly due to greater solar wind pressure.

    For the last months I have been watching the Solar Wind pressure with interest. There have been many occasions recently when the pressure indicated zero and the Solar Windspeed was shown as ~300Km/sec or less. (The metrics are shown in a sidebar on solar activity on WUWT and the front page of Solarham) We are supposed to be at or close to the maximum of Solar Cycle 24 yet these metrics are extremely low for the Solar ‘maximum’ – and I realize that use of ‘maximum’ is tends toward hyperbole for Cycle 24.

    Apart from Piers Corbyn does anyone have any idea what the impact of these anomalously low values will have?

  27. suricat says:

    gallopingcamel says: September 18, 2012 at 4:03 pm
    Responding to: “suricat, September 17, 2012 at 11:55 pm”

    Thanks for your links, they’re informative. If it doesn’t look right, there’s probably more to it.

    The main problem when observing ocean tidal amplitude is that it’s mostly out of phase with the gravitational teleconnection between the Earth and the Moon. There’s a definitive interface between the ocean and air that forces the, almost incompressible, ocean to move with/against gravity as its location phases in/out of teleconnection. Thus ‘harmonics’ plays a major role within the oceans.

    However, this isn’t the case with the atmosphere, there’s no definitive interface between the atmosphere and space, other than perhaps the ‘E/F’ layers, to observe tidal movement and certainly not at the tropopause. High atmospheric observation is also hampered by solar activity.

    The best hint of atmospheric tidal motion we can observe AFAIK relates to surface barometric data, semi-diurnal recurrence (diurnal recurrence is temperature related) and atmospheric winds associated with these perturbations. Detection would be a lot easier if the atmosphere wasn’t compressible. ;)

    Here’s a link giving the history and an outline of the subject:
    http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-810-dynamics-of-the-atmosphere-spring-2008/lecture-notes/chapter_9.pdf

    Best regards, Ray.

  28. David A says:

    The 37W/m^2 feature of the solar variation between the SH summer, and the NH summer, is educational on many levels. Despite the massive increase in radiation, dwarfing any CO2 induced GH affect, the atmosphere is cooler.

    I have often asked, does the planet, (oceans, land and atmosphere), gain energy during the SH summer, or lose energy?

    Any guesses?

  29. David A says:

    It is a serious question. Seven percent more energy in, the atmosphere cools? Increased albedo due to more land mass in N.H; increased insolation into SH dominant oceans. Climate models must accurately predict the annual oscillation if they hope to model long term changes in insolation, clouds, etc. Do the climate models accurately model the seasons?

  30. David A says:

    What would happen to the earths average T is the seasons changed, IE, the sun was reached its nearest point n the NH summer, vs the SH summer?

  31. E.M.Smith says:

    Polar vortex ought to matter along with the polar night jet. All the air coming down the swirl in the long cold dark; that rose in the warm sunny parts…

    @David A.: The ice age cycle is driven by that change interacting with tilt and orbital circularity. We only leave a glacial in one combination. That combination is now ending. Once arctic ice is persistant, the glacial returns. Summer solar at north pole is a critical element.

  32. E.M.Smith says:

    @David A.: I ought to add: It isn’t the closeness but the length of summer. With NH summer at furthest from the sun, it lasts several days longer than at perihelion. More ice melt time matters more than a bit more sun strength per day.