Atmospheric tides are global-scale periodic oscillations of the atmosphere

Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

. In many ways they are analogous to ocean tides. Atmospheric tides can be excited by:

• The regular day
  Day
  A day is a unit of time, commonly defined as an interval equal to 24 hours. It also can mean that portion of the full day during which a location is illuminated by the light of the sun...
  
  /night
  Night
  Night or nighttime is the period of time when the sun is below the horizon. This occurs after dusk. The opposite of night is day...
  
   cycle in the insolation
  Insolation
  Insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter or kilowatt-hours per square meter per day...
  
   of the atmosphere
  Atmosphere
  An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...
  
  
• The gravitational field
  Gravitational field
  The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
  
   pull of the Moon
  Moon
  The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
  
  
• Non-linear interactions between tides and planetary waves.
• Large-scale latent heat
  Latent heat
  Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...
  
   release due to deep convection
  Deep convection
  Deep convection is a convective process in a medium, usually in an atmosphere or ocean, that in some sense spans the vertical.Within the atmosphere, this means from the surface to above the 500 hPa level, generally stopping at or defining the tropopause at around 200 hPa...
  
   in the tropics
  Tropics
  The tropics is a region of the Earth surrounding the Equator. It is limited in latitude by the Tropic of Cancer in the northern hemisphere at approximately  N and the Tropic of Capricorn in the southern hemisphere at  S; these latitudes correspond to the axial tilt of the Earth...
  
  .

General characteristics

The largest-amplitude atmospheric tides are mostly generated in the troposphere

Troposphere

The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....

 and stratosphere

Stratosphere

The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

 when the atmosphere

Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

 is periodically heated as water vapour and ozone

Ozone

Ozone , or trioxygen, is a triatomic molecule, consisting of three oxygen atoms. It is an allotrope of oxygen that is much less stable than the diatomic allotrope...

 absorb solar radiation during the day. The tides generated are then able to propagate away from these source regions and ascend into the mesosphere

Mesosphere

The mesosphere is the layer of the Earth's atmosphere that is directly above the stratosphere and directly below the thermosphere. In the mesosphere temperature decreases with increasing height. The upper boundary of the mesosphere is the mesopause, which can be the coldest naturally occurring...

 and thermosphere

Thermosphere

The thermosphere is the biggest of all the layers of the Earth's atmosphere directly above the mesosphere and directly below the exosphere. Within this layer, ultraviolet radiation causes ionization. The International Space Station has a stable orbit within the middle of the thermosphere, between...

. Atmospheric tides can be measured as regular fluctuations in wind

Wind

Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

, temperature

Temperature

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

, density

Density

The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

 and pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

. Although atmospheric tides share much in common with ocean tides they have two key distinguishing features:

1. Atmospheric tides are primarily excited by the Sun
   Sun
   The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
   
   's heating of the atmosphere whereas ocean tides are excited by the Moon
   Moon
   The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
   
   's gravitational pull and to a lesser extent by the Sun's gravity. This means that most atmospheric tides have periods of oscillation
   Oscillation
   Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
   
    related to the 24-hour length of the solar day whereas ocean tides have periods of oscillation related both to the solar day as well as to the longer lunar day
   Lunar day
   In space exploration, a lunar day is the period of time it takes for the Earth's Moon to complete one full rotation on its axis with respect to the Sun. Equivalently, it is the time it takes the Moon to make one complete orbit around the Earth and come back to the same phase...
   
    (time between successive lunar transits) of about 24 hour
   Hour
   The hour is a unit of measurement of time. In modern usage, an hour comprises 60 minutes, or 3,600 seconds...
   
   s 51 minute
   Minute
   A minute is a unit of measurement of time or of angle. The minute is a unit of time equal to 1/60th of an hour or 60 seconds. In the UTC time scale, a minute on rare occasions has 59 or 61 seconds; see leap second. The minute is not an SI unit; however, it is accepted for use with SI units...
   
   s.
2. Atmospheric tides propagate in an atmosphere where density varies significantly with height
   Height
   Height is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example "The height of the building is 50 m" or "The height of the airplane is 10,000 m"...
   
   . A consequence of this is that their amplitude
   Amplitude
   Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
   
   s naturally increase exponentially as the tide ascends into progressively more rarefied regions of the atmosphere (for an explanation of this phenomenon, see below). In contrast, the density
   Density
   The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
   
    of the oceans varies only slightly with depth and so there the tides do not necessarily vary in amplitude with depth.

At ground level, atmospheric tides can be detected as regular but small oscillations in surface pressure with periods of 24 and 12 hours. However, at greater heights the amplitudes of the tides can become very large. In the mesosphere (heights of ~ 50–100 km) atmospheric tides can reach amplitudes of more than 50 m/s and are often the most significant part of the motion of the atmosphere.

The reason for this dramatic growth in amplitude from tiny fluctuations near the ground to oscillations that dominate the motion of the mesosphere lies in the fact that the density of the atmosphere decreases with increasing height. As tides or waves propagate upwards, they move into regions of lower and lower density. If the tide or wave is not dissipating, then its kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

 density must be conserved. Since the density is decreasing, the amplitude of the tide or wave increases correspondingly so that energy is conserved. The amplitude of a wave at a height of z can thus be described by the equation:

where is the initial amplitude of the wave, is height and is the scale height

Scale height

In various scientific contexts, a scale height is a distance over which a quantity decreases by a factor of e...

 of the atmosphere.

Following this growth with height atmospheric tides have much larger amplitudes in the middle and upper atmosphere than they do at ground level.

Solar atmospheric tides

The largest amplitude atmospheric tides are generated by the periodic heating of the atmosphere by the Sun – the atmosphere is heated during the day and not heated at night. This regular diurnal (daily) cycle in heating generates tides that have periods related to the solar day. It might initially be expected that this diurnal heating would give rise to tides with a period of 24 hours, corresponding to the heating's periodicity. However, observations reveal that large amplitude tides are generated with periods of 24 and 12 hours. Tides have also been observed with periods of 8 and 6 hours, although these latter tides generally have smaller amplitudes. This set of periods occurs because the solar heating of the atmosphere occurs in an approximate square wave

Square wave

A square wave is a kind of non-sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels...

 profile and so is rich in harmonics. When this pattern is decomposed into separate frequency components using a fourier transform

Fourier transform

In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

, as well as the mean and daily (24-hr) variation, significant oscillations with periods of 12, 8 and 6 hrs are produced. Tides generated by the gravitational effect of the sun are very much smaller than those generated by solar heating. Solar tides will refer to only thermal solar tides from this point.

Solar energy is absorbed throughout the atmosphere some of the most significant in this context are water vapor

Water vapor

Water vapor or water vapour , also aqueous vapor, is the gas phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Under typical atmospheric conditions, water vapor is continuously...

 at (~0–15 km) in the troposphere

Troposphere

The troposphere is the lowest portion of Earth's atmosphere. It contains approximately 80% of the atmosphere's mass and 99% of its water vapor and aerosols....

, ozone

Ozone

Ozone , or trioxygen, is a triatomic molecule, consisting of three oxygen atoms. It is an allotrope of oxygen that is much less stable than the diatomic allotrope...

 at (~30 to 60 km) in the stratosphere

Stratosphere

The stratosphere is the second major layer of Earth's atmosphere, just above the troposphere, and below the mesosphere. It is stratified in temperature, with warmer layers higher up and cooler layers farther down. This is in contrast to the troposphere near the Earth's surface, which is cooler...

 and molecular oxygen and molecular nitrogen at (~120 to 170 km) in the thermosphere

Thermosphere

The thermosphere is the biggest of all the layers of the Earth's atmosphere directly above the mesosphere and directly below the exosphere. Within this layer, ultraviolet radiation causes ionization. The International Space Station has a stable orbit within the middle of the thermosphere, between...

. Variations in the global distribution and density of these species results in changes in the amplitude of the solar tides. The tides are also affected by the environment through which they travel.

Solar tides can be separated into two components: migrating and non-migrating.

Migrating solar tides

Migrating tides are sun synchronous – from the point of view of a stationary observer on the ground they propagate westwards with the apparent motion of the sun. As the migrating tides stay fixed relative to the sun a pattern of excitation is formed that is also fixed relative to the Sun. Changes in the tide observed from a stationary viewpoint on the Earth's surface are caused by the rotation of the Earth with respect to this fixed pattern. Seasonal variations of the tides also occur as the Earth tilts relative to the Sun and so relative to the pattern of excitation.

The migrating solar tides have been extensively studied both through observations and mechanistic models.

Non-migrating solar tides

Non-migrating tides can be thought of as global-scale waves with the same periods as the migrating tides. However, non-migrating tides do not follow the apparent motion of the sun. Either they do not propagate horizontally, they propagate eastwards or they propagate westwards at a different speed to the sun. These non-migrating tides may be generated by differences in topography

Topography

Topography is the study of Earth's surface shape and features or those ofplanets, moons, and asteroids...

 with longitude, land-sea contrast and surface interactions. An important source is latent heat

Latent heat

Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...

 release due to deep convection

Deep convection

Deep convection is a convective process in a medium, usually in an atmosphere or ocean, that in some sense spans the vertical.Within the atmosphere, this means from the surface to above the 500 hPa level, generally stopping at or defining the tropopause at around 200 hPa...

 in the tropics

Tropics

The tropics is a region of the Earth surrounding the Equator. It is limited in latitude by the Tropic of Cancer in the northern hemisphere at approximately  N and the Tropic of Capricorn in the southern hemisphere at  S; these latitudes correspond to the axial tilt of the Earth...

.

The primary source for the 24-hr tide is in the lower atmosphere where surface effects are important. This is reflected in a relatively large non-migrating component seen in longitudinal differences in tidal amplitudes. Largest amplitudes have been observed over South America

South America

South America is a continent situated in the Western Hemisphere, mostly in the Southern Hemisphere, with a relatively small portion in the Northern Hemisphere. The continent is also considered a subcontinent of the Americas. It is bordered on the west by the Pacific Ocean and on the north and east...

, Africa

Africa

Africa is the world's second largest and second most populous continent, after Asia. At about 30.2 million km² including adjacent islands, it covers 6% of the Earth's total surface area and 20.4% of the total land area...

 and Australia

Australia

Australia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the mainland of the Australian continent, the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans. It is the world's sixth-largest country by total area...

.

Lunar atmospheric tides

Atmospheric tides are also produced through the gravitational effects of the Moon. Lunar (gravitational) tides are much weaker than solar (thermal) tides and are generated by the motion of the Earth's oceans (caused by the Moon) and to a lesser extent the effect of the Moon's gravitational attraction on the atmosphere.

Classical tidal theory

The basic characteristics of the atmospheric tides are described by the classical tidal theory.
By neglecting mechanical forcing and dissipation

Dissipation

In physics, dissipation embodies the concept of a dynamical system where important mechanical models, such as waves or oscillations, lose energy over time, typically from friction or turbulence. The lost energy converts into heat, which raises the temperature of the system. Such systems are called...

, the classical tidal theory assumes
that atmospheric wave
motions can be considered as linear perturbations of an initially motionless
zonal mean state that is horizontally stratified and isothermal. The two major results of the classical theory are

• atmospheric tides are eigenmodes of the atmosphere described by Hough functions

• amplitudes grow exponentially with height.

Basic equations

The primitive equations

Primitive equations

The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...

 lead to the linearized equations for perturbations (primed variables)
in a spherical isothermal atmosphere:

• horizontal momentum equations

• energy equation

• continuity equation

with the definitions

• eastward zonal wind
• northward meridional wind
• upward vertical wind
• geopotential,
• square of Brunt-Vaisala (buoyancy) frequency
• angular velocity of the Earth
• density
• altitude
• geographic longitude
• geographic latitude
• heating rate per unit mass
• radius of the Earth
• gravity acceleration
• constant scale height
• time

Separation of variables

The set of equations can be solved for atmospheric tides, i.e., longitudinally propagating waves of zonal wavenumber
and frequency . Zonal wavenumber is a positive
integer so that positive values for correspond to eastward propagating tides
and negative values to westward propagating tides. A separation approach of the form

and doing some math yields expressions for the latitudinal and vertical structure of the tides.

Laplace's tidal equation

The latitudinal structure of the tides is described by the horizontal structure equation which is also called Laplace's tidal equation:

with Laplace operator

using , and eigenvalue

Hence, atmospheric tides are eigenoscillations (eigenmodes)of Earth's atmosphere with eigenfunctions , called Hough functions, and eigenvalues . The latter define the equivalent depth which couples the latitudinal structure of the tides with their vertical structure.

Vertical structure equation

For bounded solutions and at altitudes above the forcing region, the vertical structure equation in its canonical form is:

with solution



using the definitions

• 
  
  • 
    
    • 
      
      • 
        

        Propagating solutions

        Therefore, each wavenumber/frequency pair (a tidal component) is a
        superposition of associated Hough functions (often called tidal modes in the literature)
        of index n. The nomenclature is such that a negative value of n refers to evanescent
        modes (no vertical propagation) and a positive value to propagating modes.
        The equivalent depth
        is linked to the vertical wavelength , since is the vertical wavenumber:
        

        
        
        For propagating solutions , the vertical group velocity
        

        
        
        becomes positive (upward energy propagation) only if for westward
        or if for eastward
        propagating waves.
        At a given height , the wave maximizes for
        

        
        
        For a fixed longitude , this in turn always results in downward phase progression as time
        progresses, independent of the propagation direction. This is an important result for
        the interpretation of observations: downward phase progression in time means an upward propagation of energy and therefore a tidal forcing lower in the atmosphere.
        Amplitude increases with height
        , as density decreases.
        

        Dissipation

        Damping

        Damping

        In physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...

        
         of the tides occurs primarily in the lower thermosphere region, and may be caused by turbulence

        Turbulence

        In fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...

        
         from breaking gravity wave

        Gravity wave

        In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media which has the restoring force of gravity or buoyancy....

        
        s. A similar phenomena to ocean waves breaking on a beach

        Beach

        A beach is a geological landform along the shoreline of an ocean, sea, lake or river. It usually consists of loose particles which are often composed of rock, such as sand, gravel, shingle, pebbles or cobblestones...

        
        , the energy

        Energy

        In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

        
         dissipates into the background atmosphere. Molecular diffusion

        Diffusion

        Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

        
         also becomes increasingly important at higher levels in the lower thermosphere as the mean free path

        Mean free path

        In physics, the mean free path is the average distance covered by a moving particle between successive impacts which modify its direction or energy or other particle properties.-Derivation:...

        
        increases in the rarefied atmosphere.
        

        Effects of atmospheric tide

        The tides form an important mechanism for transporting energy input into the lower atmosphere from the upper atmosphere, while dominating the dynamics of the mesosphere and lower thermosphere. Therefore, understanding the atmospheric tides is essential in understanding the atmosphere as a whole. Modeling and observations of atmospheric tides are needed in order to monitor and predict changes in the Earth's atmosphere.

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