Thermal wind equation

This alternative formulation of geostrophic equilibrium is known as the thermal wind equation it relates the vertical shear of the zonal-mean zonal wind to the horizontal gradient of the zonal-mean temperature. If the temperature decreases toward the poles (as observed in the wintertime stratosphere), then a positive vertical shear will develop so that the zonal-mean zonal wind becomes increasingly eastward with altimde. On the other hand, if the pole-ward temperature gradient is positive (as is the case in the summer stratosphere), then the zonal wind will become more westward with altitude. This is precisely what is observed, as may be ascertained by comparing the mean temperature distributions of Fig. 5 with the zonal-mean zonal wind distributions of Fig. 6. 

The thermal wind equation relates the appearance of the summer westward and winter eastward jets in the stratosphere to the temperatiue distribution, which is in turn a result of the latitudinal variation of heating due to absorption of ultraviolet radiation by ozone and large-scale motion of air parcels. 

We concentrate on the information obtained from infrared spectroscopy and radiometry, both directly and in conjunction with other data sets, such as those from visible imaging. To provide the necessary background for the subjects of this section, we first review the equations of fluid motion and the succession of approximations leading to a tractable set of equations that can be used to describe the motion of a planetary atmosphere. Eor most of the cases considered, geostrophic balance and the associated thermal wind equations play major diagnostic roles in the inference of atmospheric motions from remotely sensed temperatures. For this reason, the derivation of these relations will be discussed in some detail. Other... 

The geostrophic relations and the associated thermal wind equations can provide significant insight into the behavior of rotating atmospheres they are the lowest order approximation in a systematic development of large-scale atmospheric dynamics. In addition, these equations have been used to obtain information on atmospheric winds from remotely sensed measurements for many of the planetary atmospheres considered here. Therefore, we examine the geostrophic... 

We have developed the basic tools necessary for the application of remotely sensed data to problems in the dynamics of planetary atmospheres. The thermal wind equations are used extensively for this purpose, while the complete set of primitive equations, (9.2.20)-(9.2.24), forms the starting point for most of the other relevant approximations and models. We now turn to selected examples of applications to specific planetary atmospheres. 

We now relate the measured temperatures to the mean zonal wind. In this case it is not possible to use the geostrophic thermal wind equation (9.2.29) as an examination of Eq. (9.2.21) shows. The ratio of the second term to the first term in the brackets on the left side of the equation is of the order of the ratio of the 243-day planetary rotation period to the four-day atmospheric rotation period or 60 hence, the second term dominates. This suggests a first approximation ... 

The oscillating wind regimes depicted schematically in Fig. 16 (and from observations in Figs. 10 and 11) are accompanied by secondary circulations in the meridional plane, and by temperature anomalies in balance with the adiabatic heating and cooling associated with the secondary circulations. The magnitude of these anomalies can be estimated because, even in the tropics, the zonal-mean zonal wind and temperature are in thermal wind balance. Near the equator, Eq. (13) can be written as follows ... 

It is less straightforward to obtain information on the meridional components of the circulation from temperature fields alone. From Eq. (9.2.28) it can be seen that the meridional component of the thermal wind shear vanishes in the zonal mean, and higher order approximations are required to treat this component of the flow. However, examination of meridional cross sections, such as that shown in Fig. 9.2.1, permits some qualitative statements to be made. For this purpose an approximate form of the thermodynamic energy equations, Eq. (9.2.24), is useful. First, consider... 

Because the time scale of the anticipated atmospheric motion associated with the temperature fluctuations is only one day, the thermal wind approximation caimot be used. However, diurnal variations in the pressure and wind fields can be estimated from the observed temperature field using classical tidal theory. The basic concept of the formulation is sketched here a detailed treatment can be found in Chapman Lindzen (1970). The theory is based on a linearization of the primative equations. A motionless atmospheric reference state is assumed with temperature profile To z) and a corresponding geopotential surface 4>o(z). It is further assumed that the diabatic heating and all other quantities vary as exp[i(j < — cot)] where s is a longitudinal wavenumber and co is 2 7r/(solar day) or integer multiples thereof. The amplitudes of the time-varying, dependent variables are taken to be sufficiently small so that only terms of first order need be retained. With these assumptions. 

Thus, for opposed flow spread, the steady state thermal flame spread model appears valid. In wind-aided flame spread, it seems appropriate to modify our governing equation for the thermally thin case as... 

These equations are generalizations of the logarithmic layer equations to the case of a thermally stratified layer. We remind the reader that the adiabatic temperature profile in a stagnant layer is the familiar 1 "C/100 m decrease. However, in the presence of a mean wind in the x direction with a logarithmic profile, the neutral temperature profile is given by... 

Bottom water currents in sluggish streams (i.e., bayous), lakes, estuaries, and other near-shore marine waters are moved by the wind at the surface. Both thermal and salinity stratification in these waters is a factor influencing the magnitudes of the bottom-water transport coefficients. Although this subject of MTCs has received limited study, some estimation methods are proposed. For unstratified water bodies. Equation 12.10 is useful wind speed is a key independent variable. For stratified lakes surface winds cause seiches that generate bottom water currents. Equations 12.11 through 12.13 can be used with seiche water displacement heights. To estimate bottom currents, these values are converted to bottom friction velocities with Equation 12.8, Equation 12.1 is then used for the MTC estimate. Bed characteristics can be used as proxies for bottom currents see Table 12.5. 

In the absence of hydraulic or wind forces, the water becomes quiescenf but natural or free convection processes remains operative. Driven by bottom residing thermal or concentration gradients. Equations 12.14 and 12.15 may be used for estimating these low-end MTCs. The chemical diffusion coefficient in the porewaters of the upper sediment layer is the key to quantifying the sediment-side MTC. Use Archie s law. Equation 12.18, to correct the aqueous chemical molecular diffusivity for the presence of the bed material. Bed porosity is the key independent variable that determines the magnitude of the correction factor. See Table 12.7 for typical porosity values in sedimentary materials. Eor colloids in porewaters. Equation 12.18 applies as well. The Stokes-Einstein equation (Equation 12.19) is recommended and some reported particle Brownian diffusion coefficients appear in Tables 12.9 and 12.10. Under quasisteady-state conditions, Equation 12.23 is appropriate for estimating the bed-side MTCs. 

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