Atmospheric tides

Atmospheric tidal waves with periods of a whole and a half solar day (meridional wave numbers m = 1 and 2) and waves with periods of a whole and a half year (meridional wave number m = 0) are large-scale atmospheric waves with horizontal dimensions of the order of magnitude Circumference of the earth. For the sake of completeness, you can also add the zonal climatic mean.

introduction

The earth's atmosphere is a huge waveguide with a solid subsurface (the earth's surface) and open at the top. Any number of eigenmodes can exist in such a waveguide. However, the atmosphere is a non-linear chaotic system, so that only large-scale waves with small amplitudes can be filtered out of the meteorological noise. The smaller-scale waves contribute to the meteorological noise. Large-scale atmospheric planetary waves exist for all periods from a few hours to years. The all-day tidal wave and the annual wave clearly stand out from this broad spectrum .

Tidal waves as well as annual and semi-annual waves are generated by differential solar heating in the atmosphere. They are therefore called thermal tides. The gravitational tidal force of the moon, which is known to be of fundamental importance for ocean tides , can also force lunar atmospheric tides, which are much weaker than thermal tides. Gravitationally forced solar tides are so small that they cannot be separated from the solar thermal tides.

Solar heat source

Part of the visible solar radiation is absorbed directly in the lower atmosphere, preferably in the clouds (around 24%). This part is an essential source for the generation of thermal tides. A slightly larger part (46%) penetrates to the surface of the earth and heats the subsoil. Both clouds and the earth's surface reflect about 30% back into space ( albedo ). The ocean surface is particularly heated at low latitudes and the surface water evaporates. Moist air penetrates upwards. When clouds form, the air saturated with water vapor gives off latent heat to the environment. Due to dynamic processes, this latent heat is distributed to the middle latitudes and thereby modifies the heat generated by the directly absorbed solar radiation.

A wide band of solar ultraviolet radiation is absorbed ( ozone layer ) at altitudes between around 20 and 60 km . This represents a second heat source for the tidal waves. X-rays and extreme ultraviolet rays are almost completely absorbed above about 85 km (the thermosphere ) and thus form a third heat source for generating tidal waves. Finally, there is a fourth heat source in the aurora zones of the thermosphere. A magnetospheric electric convection field drives electric currents at altitudes around 100–150 km ( ionospheric dynamo layer ). Ohmic losses of these currents heat up the thermospheric gas in these latitudes.

The thermal effectiveness of the individual atmospheric layers increases with altitude. While it is of the order of magnitude of 0.01 to 0.1 W / kg in the lower atmosphere, it is 1 to 10 W / kg in the thermosphere. As a result of this difference, tides are only a marginal phenomenon in the troposphere , whereas in the thermosphere they are a dominant event.

Tidal theory

Wave equation

Figure 1 .: Rectangular box designed to simulate the atmospheric waveguide. It has a solid base (the ground) and is open at the top. The abscissa x replaces the longitude λ, the ordinate y replaces the latitude φ. A point source creates up and down wave modes. The downward traveling wave is reflected on the ground, so that above the source only upward traveling waves can exist (radiation condition). The drawn meridional structure of a wave is that of the all-day tidal wave (1, -2, -1).

For a rectangular waveguide of the shape in Fig. 1, the solution of the wave equation (e.g. for the vertical wind w) is:

${\displaystyle (1)\qquad w=w_{0}[e^{ik_{z}}-e^{-ik_{z}}]\cos(k_{y})e^{i(k_{x}-\omega t)}}$

with (x, y, z) the Cartesian coordinates, t the time, ω the angular frequency, k _x = 2πm / L, k _y = π (n-1/2) / B constant horizontal wave numbers, (n, m) whole Numbers and (L, 2B) the dimensions of the box. The vertical wavenumber is then determined from the dispersion equation

${\displaystyle (2)\qquad k_{z}={\sqrt {\omega ^{2}/c^{2}-k_{x}^{2}-k_{y}^{2}}}}$

with c the speed of light . Eq. (1) describes the propagation of two free, characteristic waves running upwards and downwards, which above their source area can only propagate upwards (radiation condition).

A similar equation can be developed for ultra-long waves in an isothermal spherical background atmosphere with a constant scale height H ( Laplace equation ). For this, a strict linearization of the hydrodynamic and thermodynamic equations is necessary, and the wave amplitudes must be small compared to the parameters (density, pressure, temperature) of the background atmosphere. This applies to the tidal waves in both the troposphere and the thermosphere . In the altitude range between about 50 and 100 km, their amplitudes can become so large that non-linear effects, in particular turbulence, can destroy the wave structure.

The linearization enables the variables to be separated into the horizontal structures geographical latitude and longitude (φ, λ) and height z. The upward wave of Eq. 1 equivalent solution is then

${\displaystyle (3)\qquad w=w_{0}e^{(1-ik_{z})\zeta }\Theta _{n}^{m}(\phi ,\nu )e^{i(m\lambda -\omega t)}}$

Here the length dependence is the same as in Eq. 1 (x = a λ; with a being the radius of the earth). The meridional structure is now replaced by the Hough function Θ _n ^m (φ, ν), which is derived not only from the zonal wave number n, but also from the meridional wave number m and the normalized frequency ν = ω / Ω (with Ω = 7, 29 × 10 ⁻⁵ s ^{−1 of} the sidereal angular frequency of the earth's rotation). Furthermore, ζ = z / (2H) is a normalized height. The Hough function is the solution to Laplace's equation. It can only be determined numerically and can be approximated by a sum of spherical functions . In the thermosphere these eigenmodes develop into the spherical functions themselves. As in Eq. 1, the zonal wave number n is a measure of the number of zeros within an earth quadrant. The vertical wavenumber k _z is now

${\displaystyle (4)\qquad k_{z}={\sqrt {\epsilon _{n}^{m}/\epsilon _{c}-1}}}$

with ε _n ^m an eigenvalue ( Lamb parameter) and ε _c = (aΩ) ² / (κgH) ≈ 9.5 of a constant. H ≈ 8 km is the mean scale height , g = 9.81 m / s ² the gravitational acceleration and κ = 1 - c _v / _p = 0.29 (with c _v , c _p the specific heats of the atmospheric gas at constant volume or constant pressure). The exponential increase in waves with height in Eq. 3 according to exp {z / (2H)}. As long as the waves propagate upwards without losses into an area of ​​exponentially decreasing density, their amplitudes must grow exponentially for reasons of energy conservation. The consequences of this growth are especially noticeable in the middle atmosphere, where the amplitudes can reach such magnitudes that the conditions for linearization are no longer met. The waves become turbulent and give off their wave energy to the gas surrounding them. It is also noteworthy for atmospheric wave propagation in general that the vertical components of the phase and group velocity run in opposite directions. An upward wave has a phase velocity that is directed downward.

From Eq. 4 it can be seen that the vertical wave number k _z becomes imaginary when ε _n ^m <ε _c . Waves with real values ​​of k _z are called internal waves. They have finite vertical wavelengths and can transport wave energy upwards. Their amplitudes grow exponentially with height. Waves with imaginary vertical wave numbers are called external waves. They have infinitely large vertical wavelengths, and their wave energy decreases exponentially outside their source area if ik _z > 1. They cannot transport wave energy. Atmospheric gravity waves and most of the atmospheric tides that are excited in the lower atmosphere are internal waves. Since their amplitudes grow exponentially, these waves are destroyed by turbulence in the height range around 100 km at the latest. All seasonal waves (m = 0) are external waves.

Solution of Laplace's equation

Michael Longuet-Higgins completely solved the Laplace equation . He discovered that there are also negative eigenvalues ​​ε _n ^m . There are two classes of wave modes: Class I waves, which are marked with positive indices n, and Class II waves with negative n. Class II waves only exist due to the Coriolis force and disappear for periods shorter than 12 h (| ν |> 2). Waves with even n are symmetric with respect to the equator, waves with odd n are antisymmetric. The waves are identified by the number triple (m, n, ν).

Figure 2. Meridional structure of the pressure amplitudes of the Hough functions of all-day waves (m = 1; ν = -1) (left) and half-day waves (m = 2; ν = -2) (right) in the northern hemisphere. Solid lines: symmetrical waves; dashed lines: antisymmetric waves

The fundamental all day tidal wave (1, -2, -1) is an external wave. It moves west with the sun. It has the eigenvalue of ε _-2 ¹ = - 12.56. The most important half-day wave (2, 2, -2) is an internal wave with ε ₂ ¹ = 12.21. Fig. 2 shows the meridional structures of the pressure amplitudes of the Hough functions of the most important all-day (m = 1; left) and half-day waves (m = 2; right). The all-day tidal wave (1, -2.1), which is best adapted to the heat source in the troposphere in terms of its meridional structure , plays only a marginal role as an external wave in the lower atmosphere. In the thermosphere , however, this wave develops into the dominant tidal wave. It drives the electric Sq current at altitudes between about 100 and 200 km ( ionospheric dynamo layer ).

Loss processes

For a linearization of the hydrodynamic and thermodynamic equations, a parameterization of the friction and heat losses is introduced. The turbulent friction is replaced by a Rayleigh friction term ν _R , proportional to the negative horizontal wind U , and the turbulent heat conduction by a Newtonian cooling term ν _N , proportional to the negative temperature. Both terms are purely empirical numbers that must be derived from the observations. The loss processes now appear in the complex (normalized) frequencies ν _r = ν + iν _R and ν _h = ν + ν _N , so that the vertical wave number k _z becomes complex:

${\displaystyle (5)\qquad k_{z}={\sqrt {\epsilon _{n}^{m}\nu _{r}/(\epsilon _{c}\nu _{h})-1}}}$

The ratio ν _R / ν _N is called the Prandtl number . The terms ν _R and ν _N in the troposphere have a magnitude of approx. 0.1 (or reciprocal values ​​of 1 / ν _R ≈ 10 days) and are therefore insignificant for tides with periods of 24 h and less. In the thermosphere , however, due to the collision of the neutral gas with the ionospheric plasma, they reach values ​​of approx. 1 and greater (or reciprocal values ​​of 1 / ν _R <1 day). This means that all tidal waves above 150 km become external waves and that their Hough functions gradually degenerate into spherical functions . The mode (1, -2, -1) becomes the spherical function P ₁ ¹ (φ), mode (2, 2, -2) develops into P ₂ ² (φ) etc., with φ the geographical latitude. In the thermosphere, the all-day mode (1, -2, -1) is the dominant tidal wave with temperature amplitudes of approx. 15% of the globally averaged exosphere temperature (approx. 1000–1500 K) and horizontal winds of the order of 100 m / s ( see Fig. 4).

Vertical structure

Due to the linearization of the hydrodynamic and thermodynamic equations, the separation into horizontal and vertical structures is possible. The Laplace equation provides the horizontal structure in the form of Hough's wave modes. The eigenvalue ε _n ^m in Eq. 4 connects the horizontal structure with the equations of the vertical structure, which are only dependent on the height z. There are particularly simple solutions for an isothermal atmosphere ( scale height H = const.). The solution consists of a particulate solution that is directly proportional to the heat source and a free upward wave with the vertical wave number k _z in Eq. 4. The upper boundary condition is the radiation condition: Above the heat source only waves running upwards may exist.

Observations

Solar thermal tidal waves (m> 0)

Wandering waves

The two main solar thermal tidal waves are the all-day and half-day tidal waves that travel west with the sun. Their pressure amplitudes averaged over a year on the ground have the form (in Pa)

${\displaystyle S_{1}=59\cos ^{3}\phi \cos[\tau _{S}-5,2h]}$

${\displaystyle S_{2}=116\cos ^{3}\phi \cos[2(\tau _{S}-9,7h)]}$

with τ _S = Ω _S t + λ the local time, Ω _S the angular frequency of a solar day, (φ, λ) geographical latitude and longitude and t the universal time. Both sources are symmetrical about the equator. Wave S ₁ consists of the fundamental external mode (1, -2, -1) and the next higher internal mode (1, 2, -1) (see Fig. 2). The wave S ₂ essentially corresponds to the internal mode (2, 2, -2). The amplitude of the all-day wave is smaller by a factor of 2 than that of the half-day wave, although the spectral amplitude of its heat source is larger by a factor of 2. The external wave is thus suppressed by a factor of 4 compared to the internal wave.

Not wandering waves

These are large-scale standing waves with a period of one day or one year and harmonics. They depend on the universal time and are generated by orographically determined differences in the earth's surface (continent-ocean contrast, topographical differences, climatological differences, etc.). An important source of these waves is convection in the tropics.

The standing antisymmetric tidal wave with wave number m = 1 and a period of one year is of particular importance. It is created by pressure differences along a degree of latitude. In winter there is a high pressure area over the northern Atlantic and a low pressure area over Siberia with temperature differences of around 50 ° C. In summer it's the other way around. Such a wave can be described by the Rossby-Haurwitz wave Θ _-3 ¹ (for Rossby-Haurwitz waves ε = 0 applies). Your zonal wind has approximately the width structure of the spherical function P ₂ ¹ = 3 sin φ cos φ.

With the help of such an excitation function, the seasonal component of the polar movement of the geographic pole in relation to the rotation axis of the earth's body can be determined from Euler's equations of the gyroscopic movements . The pressure amplitude of the excitation function is of the size 1.2 hPa. The same wave type with the period from approx. 430 to 440 sidereal days is responsible for the Chandler wobbling in the vicinity of the Chandler resonance period of 441 sidereal days.

Lunar tidal waves

The strongest lunar tidal wave has the pressure amplitude on the earth's surface (in Pa) of

${\displaystyle L_{2}=6\cos ^{3}\phi \cos[2\tau _{L}-15^{o}]}$

with τ _L the lunar local time. This amplitude on the earth's surface is 20 times smaller than that of the solar half-day wave. It is difficult to filter them out of the meteorological noise. Here, too, there are smaller harmonics.

Seasonal waves (m = 0)

Lower atmosphere

Figure 3 .: Climatic mean of the zonal wind as a function of height and width; the east wind zone near the equator is dotted.

The climatic mean of the zonal wind (m = ν = 0) within the troposphere shows in middle and higher latitudes mainly westerly winds (directed from west to east), which increase with height and at about 30 ° to 45 ° latitude with two jet streams culminate at a height of 12 to 15 km with strengths of the order of 20 to 30 m / s (see Fig. 3). During the solstices, the jet current is increased by around 10 m / s in the summer hemisphere and weakened accordingly in the winter hemisphere. At low latitudes, an easterly wind (the trade winds ) overlaps , which has a maximum on the ground of about 5 m / s and has disappeared in about 15 km (the tropopause ).

The excess solar heat at low latitudes creates a temperature drop from the equator to the poles. In an ideal atmosphere without loss processes, the Coriolis force would generate a zonal westerly wind ( geostrophic wind ) on both hemispheres, and heat would ultimately be transported by conduction. The real atmosphere, however, is unstable, and the heat and pressure balance within the west wind drift is provided by turbulence cells of all sizes from high and low pressure areas to small turbulence. The heat transport from low to high latitudes is extremely complex ( baroclinic instability). In our approximation, this is taken into account across the board by the loss terms Rayleigh friction ν _R and Newton cooling ν _N. For the daily synoptic weather events, the Rossby-Haurwitz waves migrating westward relative to the westward drift are preferably involved with the meridional wave numbers m = 4 - 7. They are already averaged out in the climatic mean.

The fundamental wave mode (0, -2, iν _R ) is an external mode and has the zonal wind component

${\displaystyle u_{-2}^{0}(\phi ,z)=a_{1}(z)P_{1}^{1}(\phi )+a_{3}(z)P_{3}^{1}(\phi )+...}$

The spherical function P ₁ ¹ = cos φ describes the rigid super rotation (or retrograde rotation) of the atmospheric layer at the height z. On the earth's surface, however, there is friction between atmospheric gas and earth. The westerly wind of the wave mode (0, -2, iν _R ) therefore accelerates the earth's rotation. The reaction time is about 7 days, so that the earth's body does not react to small periodic processes. However, a fundamental law of physics says that the angular momentum of a system remains constant as long as no external torques are effective. In the climatic mean, the earth must not be accelerated (or decelerated) by the atmosphere, and the rigid rotation term of the atmosphere on the earth's surface has to disappear on the mean, so that the atmosphere and the earth's body are decoupled in the climatic mean. However, this means that an easterly wind term u _S = - a _S  cos φ must be added to the wave mode (0, -2, iν _R ) , which forces this decoupling (a ₁ + a _S = 0 on the earth's surface z = 0) and is responsible for the trade winds. The wave mode that describes the rigid superrotation of the atmosphere is a Rossby -Haurwitz wave with the meridional wave number m = 0.

A simple model of the zonal wind with only the wave mode (0, -2, iν _R ) and a Rossby-Haurwitz wave can be constructed, with the disappearance of the super-rotation component of the zonal wind and a plausible heat source with a maximum in 5 km altitude can be assumed. The result (Fig. 3) already shows the basic structure of the observations. The position of the beam current at 60 ° instead of 45 ° latitude is due to the simple model and can easily be corrected with other wave modes. The zonal wind near the earth's surface corresponds to a first approximation of the spherical function

${\displaystyle P_{3}^{1}(\phi )=1.5\cos(\phi )(5\sin ^{2}(\phi )-1)}$

It describes a westerly wind in middle latitudes and an easterly wind in low latitudes. The zero point of this spherical function is at φ = ± 27 °. On average, the atmosphere in middle and high latitudes gives off angular momentum to the earth's body, which is added back to the atmosphere by the earth's body in lower latitudes. This also applies to all other wind components with n> 1. Averaged over the sphere, its angular momentum transmission disappears.

This model is deliberately simplified and assumes a homogeneous surface of the earth in order to represent the essential processes. The real surface structure of the earth will certainly modify the picture.

Conservation of angular momentum

Because of the conservation of angular momentum, a temporary change in the atmospheric angular momentum must become noticeable in a corresponding change in the angular momentum of the earth's body. This is in fact observed in the year-round and half-yearly amplitudes of the zonal wind, which are correlated with corresponding fluctuations in the length of the day. The periodic change in the amplitude of an equivalent rigid rotation of the atmosphere (ie the entire atmosphere) of Δa ₁ ≃ 0.9 m / s corresponds to a change in day length of Δτ ≃ 34 milliseconds, with a maximum on February 3rd. Corresponding figures for the six-month period are Δa ₁ ≃ 0.8 m / s and 0.29 milliseconds with maxima on May 8th and November 8th. There are also fluctuations of 10 days on the order of 0.1 milliseconds, as well as fluctuations that reflect the El Niño phenomenon in the trade winds over the Pacific.

Medium atmosphere

The dominant wave in the middle atmosphere is the year-round wave (0, -1, iν _R ) (ν << ν _R ). During the equinoxes, two jet streams form: a westerly wind jet stream on the winter hemisphere and an east wind jet stream on the summer hemisphere, each 70 m / s at a height of about 65 to 70 km and a latitude of 50 ° to 60 °. The climatic mean and the half-yearly wave, on the other hand, are relatively small and of the order of 20 m / s. The heat source for these beam currents is the absorbed solar ultraviolet radiation, which also creates the ozone layer .

Tides in the thermosphere

Above about 150 km all atmospheric waves degenerate into external waves, and a vertical wave structure is hardly visible any more. Their meridional structure is that of the spherical functions P _n ^m with n a zonal wave number and m the meridional wave number (m = 0: zonally averaged waves; m = 1: all-day waves; m = 2: half-day waves, etc.). In a first approximation, the thermosphere behaves like a damped oscillator system with a low-pass filter effect. This means that small-scale waves (with large wave numbers n and m) are suppressed compared to the large-scale waves. In the case of low magnetospheric activity, the observed temporally and spatially varying exosphere temperature can be described by a sum of spherical functions:

${\displaystyle (6)\qquad T(\phi ,\lambda ,t)=T_{oo}[1+\Delta T_{2}^{0}P_{2}^{0}(\phi )+\Delta T_{1}^{0}P_{1}^{0}(\phi )\cos[\omega _{a}(t-t_{a}]+\Delta T_{1}^{1}P_{1}^{1}(\phi )\cos(\tau -\tau _{d})+...]}$

Figure 4.Schematic meridional-height cross-section of the circulation systems of (a) symmetrical wind component of the zonal mean (P ₂ ⁰ ), of (b) antisymmetrical wind component (P ₁ ⁰ ), and (d) symmetrical all-day wind component (P ₁ ¹ ) at 3 am and 3 pm local time. (c) shows the horizontal wind vectors of the all day wave in the Northern Hemisphere

It is φ the geographical latitude, λ the geographical longitude, t the time, ω _a the angular frequency of the annual period, τ = ω _d t + λ the local time and ω _d the angular frequency of a solar day. t _a = June 21 is the time of the beginning of summer in the northern hemisphere and τ _d = 15:00 is the local time of the maximum temperature.

The first term on the right in Eq. 6 is the global mean temperature of the exosphere (of the order of T _∞ ≈ 1000 K). The second term [(with P ₂ ⁰ = 0.5 (3 sin ² φ - 1)] is generated by the different solar heating in low and high latitudes. A thermal wind system is created with winds towards the poles in the upper circulation branch and opposite Winding in the lower branch (Fig. 4a). It ensures a heat balance between low and high latitudes. The coefficient ΔT ₂ ⁰ ≈ 0.004 is small, as the Joule warming in the aurora zones partially compensates for the solar XUV-related excess heat in low latitudes. The third term (with P ₁ ⁰ = sin φ) is responsible for the transport of the excess heat from the summer hemisphere to the winter hemisphere. Its relative amplitude is approximately ΔT ₁ ⁰ ≃ 0.13. Finally, the fourth term (with P ₁ ¹ = cos φ of the dominant tidal wave (1, -2, -1)) describes the transport of the excess heat from the day side to the night side (Fig. 4d). Its relative amplitude is approximately ΔT ₁ ¹ ≃ 0.15. Other terms (e.g. . half year, hal b-day waves etc.) have to be used for Eq. 10 can be added. However, they are of lesser importance. Corresponding sums can be derived for air pressure, air density, gas constituents, etc.

Individual evidence

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2. ^ F. Möller: Introduction to Meteorology. Bibliographical Institute, Mannheim 1973.
3. ↑ JM Forbes: Atmospheric tides I + II. In: J. Geophys. Res. 87, 1980, pp. 5222, 5241.
4. ↑ RS Stolarski, PB Hays, RG Roble: Atmospheric Heating by Solar EUV Radiation . In: Journal of Geophysical Research . tape 80 , no. 16 , 1975, p. 2266-2276 , doi : 10.1029 / JA080i016p02266 .
5. ↑ ^{a b c} S. Chapman, RS Lindzen: Atmospheric Tides. D. Reidel, Dordrecht 1970.
6. ↑ ^{a b c d} H. Volland: Atmospheric Tidal and Planetary Waves. Kluwer Publ., Dordrecht 1988.
7. ^ MS Longuet-Higgins: The eigenfunctions of Laplace's equations over a sphere. In: Phil. Trans. Roy. Soc. London, A262, 1968, p. 511.
8. ↑ JR Holton, WM Wehrbein: A numerical model of the zonal mean circulation of the middle atmosphere. In: Pageoph. 118, 1980, p. 284.
9. ↑ H. Kohl, JW King In: J. Atm. Terr. Phys. 29, 1967, p. 1045.
10. ↑ Manfred Siebert: Atmospheric Tides . In: Advances in Geophysics . Volume 7. Elsevier, 1961, ISSN  0065-2687 , pp. 105-187 .
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12. ^ ^{A b} R. J. Murgatroyd: The structure and dynamics of the stratosphere. In: GA Coby (Ed.): The Global Circulation of the Atmosphere. In: Roy. Met. Soc. London, 1969, p. 159.
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14. ^ JR Holton: An Introduction to Dynamic Meteorology. Academic Press, New York 1992.
15. ^ DS Robertson: Geophysical applications of Very-Long-Baseline Interferometry. In: Rev. Modern Phys. 14, 1993, p. 1.
16. ↑ TM Eubanks, JA Steppe, JO Dickey, PS Challahan: A spectral analysis of earths angular momentum budget. In: J. Geophys. Res. 90, 1985, p. 53787.
17. ^ RD Rosen: The axial momentum balance of earth and its fluid envelop. In: Surv. Geophys. 14, 1993, p. 1.
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19. ^ R. Hide, JO Dickey: Earth's variable rotation. In: Sciences. 235, 1991, p. 629.
20. ^ W. Köhnlein: A model of thermospheric temperature and composition . In: Planetary and Space Science . tape 28 , no. 3 , March 1980, p. 225-243 , doi : 10.1016 / 0032-0633 (80) 90015-X .
21. ↑ U. von Zahn, W. Köhnlein, KH Fricke, U. Laux, H. Trinks, H. Volland: Esro 4 model of global thermospheric composition and temperatures during times of low solar activity . In: Geophysical Research Letters . tape 4 , no. 1 , 1977, pp. 33-36 , doi : 10.1029 / GL004i001p00033 .