# 4.1 Momentum equation of the neutral atmosphere

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1 Chapter 4 Dynamics of the neutral atmosphere 4.1 Momentum equation of the neutral atmosphere Since we are going to discuss the motion of the atmosphere of a rotating planet, it is convenient to express the momentum equation of the neutral atmosphere in a reference frame rotating with the planet at an angular velocity Ω. Using the results in Section 2.1, the momentum equation of neutral gas in the rotating frame can be written as [ ] u n n m n + (u )u = m n n n g p n n m n ν ni (u v i ) 2m n n n Ω u. (4.1) t Here 2m n n n Ω v is the Coriolis force density. Here the centrifugal force is included in g. Collisions with electrons are neglected. A more complete equation would contain a frictional force density due to the viscosity of the air as well as tidal force densities due to the Moon and the Sun. 4.2 Thermospheric neutral winds Sophisticated 3-dmensional numerical models called the Thermosphere and Ionosphere General Circulation Models (TIGCMs) are now in existence, which solve the coupled continuity, momentum and energy equations for the terrestrial thermosphere, mesosphere and ionosphere. Figure 4.1a shows results from the model. The temperature distribution is rather symmetric around equator and the maximum temperature at about 15 LT is sensitive to the solar cycle variations and to magnetospheric disturbances. This temperature distribution leads to a daytime expansion of the atmosphere which is called the diurnal bulge. The horizontal pressure gradients around this diurnal bulge provide the driving force for the thermospheric winds. From the temperature distribution it is possible to derive the large scale pressure gradients and from those to calculate the wind pattern. The velocities are directed from the hot dayside across the pole to the nightside, i.e. winds are very close to being perpendicular to isotherms in the F region. 85

2 86 CHAPTER 4. DYNAMICS OF THE NEUTRAL ATMOSPHERE Figure 4.1: Neutral temperatures (isocontours) and velocities (vectors) at a constant pressure height, corresponding to an altitude of about 286 km, calculated by a numerical model (top) and an empirical MSIS model (bottom) (Roble et al., 1988).

3 4.2. THERMOSPHERIC NEUTRAL WINDS 87 Over the last couple of decades, a great deal of observational data on terrestrial thermospheric temperatures and composition have been gathered by satellite-borne neutral mass spectrometers and ground-based incoherent scatter radars. The results have been used to obtain an empirical model of the thermosphere called MSIS, which gives temperature and composition values as a function of altitude, geographic location, and geomagnetic conditions. In Fig. 4.1b the MSIS model values are shown, which are rather close to the TIGCM model results in Fig. 4.1a. The simplest mathematical formulation neglects all other terms than the pressure gradient force and the force due to collisions between neutrals and ions in eq. (4.1) and assumes that the ions are stationary (v i = 0). Then 1 u = p = k B T, (4.2) n n m n ν ni m n ν ni where the ideal gas law has been applied and the gradient in neutral density has been neglected. In this simplified situation the neutral wind should blow in a direction opposite to the temperature gradients as indicated in Fig 4.1. However, the approximation made would hold only for such large collision frequencies that the Coriolis force density could be neglected in eq. (4.1). A more realistic treatment makes the equation of motion for the neutrals far more difficult to solve, especially since the ions are not stationary when they are acted on by external electric fields. Furthermore, neutral air motion at one height can create ion motion, which in turn can set up polarization electric fields that can propagate to other heights and latitudes along the magnetic field lines. These electric fields can then affect the ion motion elsewhere. Collisions between ions and neutrals will always be present forcing the different species to drag each other. We notice that, together with the momentum equation of the neutral gas, we should solve the ion momentum equation [ ] vi n i m i t + (v i )v i = e(e + v i B) + n i m i ν in (v i u). (4.3) The Coriolis force, gravity and pressure gradient are neglected here, because they are small in comparison with the other forces. Momentum transfer between neutrals and ions is controlled by the collision frequency terms. Due to the large neutral mass density, it is the neutral gas which mainly controls the motion of the ion gas. However, the electric field term may struggle against the frictional term in eq. (4.3) and thus gradually transfer momentum from the ion gas to the neutral gas. Therefore, if the electric field is strong and has a long time for momentum transfer, the dilute ion gas can affect the neutral motion. As a matter of fact, the high electric fields at high latitudes can indeed modify the neutral gas motion. In the auroral zone we get a first approximation of the neutral wind by not neglecting the ion velocity in the collision term. Then eq. (4.2) will be modified as 1 u = p + v i = k B T + v i. (4.4) n n m n ν ni m n ν ni

4 88 CHAPTER 4. DYNAMICS OF THE NEUTRAL ATMOSPHERE Figure 4.2: Calculated contours of mass flow in g/s. Left: Solar heating as the only driving force. Right: Joule heating and momentum source also included (Dickinson et al., 1975) This implies that the ion velocity can have a strong effect on the neutral velocity. Fig. 4.2 shows the effect of magnetospheric disturbances on the thermospheric circulation of neutral air. In the left panel the meridional circulation for quiet conditions is mainly forced by solar heating which produces a large Hadley cell with warm air raising at the equator, flowing towards the poles and sinking down at high latitudes. Above the polar regions ( > 70 latitude), however, a small cell with the opposite flow direction is set up above 300 km due to heat influx from the magnetosphere outside of the plasmapause. In the right panel a large magnetic storm takes place and the opposite cell expands to lower latitudes and heights. Air rich in molecular constituents is rising up at high latitudes. This indicates that large variations, not caused by solar heating can indeed appear in neutral air motion. Such variations can be due to heating of the atmosphere by means of particle precipitation and ionospheric currents and also due to the collision term in the momentum equation. In addition to the large-scale motions, smaller scale variations due to local heating will also appear. These appear in terms of atmospheric waves, which will be discussed later. 4.3 Drag effects The effect of ions on the motion of neutral gas is called the ion drag and, conversely, neutral molecules affect the ion motion by means of the neutral drag. The momentum equations readily show that the time constant of ion drag τ in is the inverse of the ion-neutral collision frequency and the time constant of the neutral drag τ ni is the inverse of the neutral-ion collision frequency. According to eq. (2.22), τ ni τ in = ν in ν ni = m nn n m i n i. (4.5)

5 4.3. DRAG EFFECTS 89 a) u E v i0 B I v i b) B v i I v ih v iv u 0 c) ucosi v i0 v B I = ih u Figure 4.3: Drag effects. Magnetic field points downward and northward, i.e. the figure is in the magnetic meridional plane. (a) Eastward electric field, (b) southward neutral wind, and (c) field-aligned diffusion assumed. For equal neutral and ion masses this ratio is equal to the ratio of the number densities. At 250 km, the number density of the neutrals may be of the order of m 3 and that of the ions of the order of m 3, which gives a ratio for the time constants. In the daytime, close to the F layer peak, τ in may be of the order of 1 hour, whereas in the nighttime it is significantly longer, of the order of 5 hours. Hence, if an external force e.g. by an electric field sustains a constant ion velocity, it has to be acting hours to bring the neutrals in motion along with the ions. Here we demonstrate the drag effects in the F region by means of a some simplified examples. We first assume that, initially, the neutral velocity u = 0. In F region we switch on an electric field E perpendicular to the geomagnetic field. Since the ion-neutral collision frequency is small in the F region, eqs. (3.16) and (3.17) give approximately k ip = 0 and k ih = 1/(eB). This leads to eq. (3.29) and initially the ion velocity is in the Hall direction (i.e. perpendicular to the electric field and to the magnetic field) v i0 = E B B 2. (4.6) Although this equation is approximately valid, some collisions between ions and neutral still exist. Then collisions with ions can gradually put the neutral atmosphere in motion. This takes place slowly because the mass density of the neutral atmosphere is much larger than the density of the ion gas, and also because of infrequent collisions. The gravity term and pressure gradient terms in the momentum equation of the neutral gas normally balance and large forces are needed to affect this hydrostatic equilibrium. Therefore, although the ions can within a long time change the neutral velocity, the neutrals will finally move horizontally, and due to the neutral drag, they will also push the ions into a horizontal motion. We first consider the case in Fig. 4.3a, where the electric field points eastwards. Then according to eq. (4.6) v i0 points upward and northward in the magnetic meridian plane. The neutrals will be affected by the ion drag but they only accept horizontal motion. Therefore the neutral motion will be affected by the horizontal component of the ion velocity and the neutral velocity will slowly change. Simultaneously, the ions experience the neutral drag, and they will gain a field-aligned velocity component v i. The field-aligned ion velocity follows the changes of neutral

6 90 CHAPTER 4. DYNAMICS OF THE NEUTRAL ATMOSPHERE velocity quickly, because the ion mass density is small. Finally, a stationary state will be reached, and neutrals an ions will move horizontally at the same speed. The driving force in this process is due to the electric field, but the neutral air does not remain passive; it also affects the final velocity of the ions. The final neutral velocity is u = v i0 sin I = E B sin I. (4.7) In the second example ions are initially at rest and a neutral wind u 0 is blowing horizontally southwards (Fig. 4.3b). No electric field is assumed so that v i = k i (m i ν i u). When ν i is small, the Pedersen mobility is approximately zero. Then the neutral wind will cause only Hall and field-aligned motion. The Hall motion is zonal (perpendicular to the magnetic meridional plane). The component of the wind parallel to the magnetic field will drag ions until The ions therefore attain a vertical velocity given by and a horizontal velocity v i = u 0 cos I. (4.8) v iv = u 0 cos I sin I (4.9) v ih = u 0 cos 2 I. (4.10) This equilibrium is reached gained much more quickly than that in Fig. 4.3a. In the third example (Fig. 4.3c) we assume that the neutrals are originally at rest, no external electric field exist, and the plasma is diffusing along the magnetic field lines at a velocity v i0 due to gravity and pressure gradient. This has a horizontal component v ih = v i0 cos I. (4.11) The ion motion tends to accelerate the neutral horizontally to this velocity. The ions, however, also experience a neutral drag due to this motion. Thus, when the final horizontal neutral velocity is u, the neutral drag will cause an additional fieldaligned ion velocity u cos I (the Pedersen velocity is again negligible and the Hall velocity perpendicular to the magnetic meridional plane). Equilibrium is obtained when the ions and neutrals move horizontally at the same speed, i.e. This gives a horizontal neutral velocity The field-aligned ion velocity will be (v i0 + u cos I) cos I = u. (4.12) u = v i0 cos I sin 2 I. (4.13) v i = v i0 + u cos I = v i0 + v i0 cos 2 I sin 2 I = v i0 sin 2 I. (4.14)

7 4.4. E REGION NEUTRAL WINDS 91 On should notice that, in the case of Fig. 4.3a, both neutrals and ions will finally move horizontally, while in the other two cases only neutrals will move horizontally. This is because the external electric field causes field-perpendicular ion motion in the magnetic meridional plane, but horizontal neutral wind in the magnetic meridional plane cannot do the same. This situation would completely change, if the neutral wind would have a horizontal zonal component. 4.4 E region neutral winds At the E region altitudes the Coriolis term becomes more important than in the F region. In the F region, neutral winds are mainly directed perpendicular to the isobars (isoterms), but in the E region the Coriolis term turns the neutral flow direction close to the isobars (this is also the case low down in the troposphere, as seen in weather maps). However, also in the E region the wind blows from the dayside over the pole to the nightside. The E region wind speeds are only a third of the F region speeds on the average. Due to the strong coupling between the neutrals and the ions in the E region, observations of the ion motion can sometimes be used as a tracer for the neutral motion. At a steady state the neutral velocity can be solved from eq. (4.3) and the result is u = v i q m i ν in v i B q m i ν in E. (4.15) Hence, if the ion velocity and electric field is measured, the neutral wind velocity can be calculated. At high latitudes, where the magnetic field is almost vertical and the electric field is perpendicular to the magnetic field, the last two terms contribute only to the horizontal component of the neutral velocity and the vertical component is equal to the vertical component of the ion velocity. When ν in is large enough, the neutral velocity is equal to the ion velocity. For smaller values of ν in the whole eq. (4.15) has to be used. Ion velocities in the E and F regions can be measured almost simultaneously using incoherent scatter radars such as the EISCAT UHF radar at Tromsø. In the F region, where the collision term is negligble compared to the electric field and Lorentz force terms, the stationary ion momentum equation readily gives E = v i B. (4.16) Thus measuring the ion velocity high up in the F region can be used in calculating the electric field. Because of the high field-aligned mobilities of ions and electrons, no essential potential drop is possible along the field line from the F to the E region, and therefore the electric field measured at F region altitudes is also valid in the E region. This field can be inserted in eq. (4.15) and then it, together with the ion velocity measured in E region, gives the neutral wind velocity in the E region. Observations indicate that large height variations in neutral wind velocity occur at E region altitudes. These winds are due to atmospheric tides and heating caused by energy input to auroral latitudes.

8 92 CHAPTER 4. DYNAMICS OF THE NEUTRAL ATMOSPHERE Figure 4.4: Two antisymmetric and symmetric modes for the semidiurnal tide (Evans, 1976). 4.5 Tidal oscillations In a global scale, the main driving force causing the neutral air motion is due to pressure gradients. The pressure gradients are mainly caused by atmospheric heating which is a result of solar radiation. Somewhat confusingly, the resulting neutral air motions are called atmospheric tides, although they are not of gravitational origin. A more specific term would be thermal tide. Of course, gravitational tides caused by the moon and the sun are also present in the atmosphere in the same way as in oceans and the crust of the Earth, but they are less important than the thermal tides. The theory of atmospheric tides involves the solution of a set of equations governing the air motion. In addition to the momentum equation (2.17) and the continuity equation (2.19), an energy equation is needed which controls the energy input from the solar radiation and the energy transfer in the atmosphere. The solution is carried out in spherical coordinates and it involves a sum of series consisting of terms containing radial, azimuthal (longitudinal) and latitudinal parts. Because the azimuthal part is necessary periodic, it can be constructed from sinusoidal functions. The latitudinal part can be constructed from associated Legendre functions (or other special functions, which have a relation with Legendre functions). The total tide can be separated into solar and lunar tides; the latter one is purely gravitational. Tidal oscillations are periodic in azimuth and the possible periods are

9 4.5. TIDAL OSCILLATIONS 93 fractions of the length of the solar (24 h) or lunar (24.8 h) day (sinusoidal parts in the series construction). The fundamental period of the solar tide is 24 hours (azimuthal wave number 1), the next one is 12 hours (wave number 2), then 8 hours, etc. The oscillations in the solar tide travel westward with respect to the Earth s surface so that the tide remains synchronized with the position of the Sun. Different latitudinal functions are possible for each of these waves, and therefore the modes of the tide are expressed in terms of two parameters (m, n) where m is the number of cycles per day and n m is the number of nodes between the poles (not counting those at the poles). A certain semidiurnal mode (12 h period), for instance, is called S(2,4). It has a period of 12 hours and two nodes, one at each side of the equator (Fig. 4.4). In the upper mesosphere the semidiurnal tide dominates, and the diurnal solar tide dominates in the thermosphere above 250 km. Between 100 and 250 km both the diurnal and semidiurnal components can be present. The energy density of a tidal oscillation is ρu 2 /2 and thus u ρ 1/2. Therefore the neutral wind velocity tends to increase with height. This is know as tidal amplification. When the tidal amplification becomes so large that the linear theory breaks down, then also the tidal wave breaks and its energy is degraded into smaller scale motion. In addition to tides, the atmosphere contains other variations. Examples are planetary waves with periods of several days and gravity waves. Gravity waves are oscillations with periods of the order of tens of minutes. They are generated by breaking of tidal waves or local energy input like particle precipitation or sudden heating due to ionospheric currents. In the troposphere, gravity waves can be caused by a wind blowing over mountains. These are called lee waves. This mechanism is analogous to the production of ordinary sound e.g. in a flute. The wave fronts of such low altitude gravity waves may become visible in clouds, which consist of parallel stripes at constant distances.

10 94 CHAPTER 4. DYNAMICS OF THE NEUTRAL ATMOSPHERE

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