The basic characteristics of the atmospheric tides are described by the
classical tidal theory. By neglecting
mechanical forcing and
dissipation, 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 lead to the linearized equations for perturbations (primed variables) in a spherical isothermal atmosphere: {{unordered list \begin{array}{rrl} \frac{\partial u'}{\partial t} \, - \, 2 \Omega \sin \varphi \, v' \, + & \frac{1}{a \, \cos \varphi} \, \frac{\partial \Phi'}{\partial \lambda} & = 0 \\ \frac{\partial v'}{\partial t} \, + \, 2 \Omega \sin \varphi \, u' \, + & \frac{1}{a} \, \frac{\partial \Phi'}{\partial \varphi} & = 0 \end{array} \frac{\partial^2}{\partial t \partial z} \Phi' \, + \, N^2 w' = \frac{\kappa J'}{H} \frac{1}{a \, \cos \varphi} \, \left( \frac{\partial u'}{\partial \lambda} \, + \, \frac{\partial}{\partial \varphi} (v' \, \cos \varphi) \right) \, + \, \frac{1}{\varrho_o} \, \frac{\partial}{\partial z} (\varrho_o w') = 0 }} with the definitions • u eastward zonal wind • v northward meridional wind • w upward vertical wind • \Phi geopotential, \int g(z,\varphi) \, dz • N^2 square of Brunt-Vaisala (buoyancy) frequency • \Omega angular velocity of the Earth • \varrho_o density \propto \exp(-z/H) • z altitude • \lambda geographic longitude • \varphi geographic latitude • J heating rate per unit mass • a radius of the Earth • g gravity acceleration • H constant scale height • t time
Separation of variables The set of equations can be solved for
atmospheric tides, i.e., longitudinally propagating waves of zonal wavenumber s and frequency \sigma. Zonal wavenumber s is a positive integer so that positive values for \sigma correspond to eastward propagating tides and negative values to westward propagating tides. A separation approach of the form \begin{align} \Phi'(\varphi, \lambda, z, t) &= \hat{\Phi}(\varphi,z) \, e^{i(s\lambda - \sigma t)} \\ \hat{\Phi}(\varphi,z) &= \sum_n \Theta_n (\varphi) \, G_n(z) \end{align} and doing some manipulations 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'': {L} {\Theta}_n + \varepsilon_n {\Theta}_n = 0 with
Laplace operator {L}=\frac{\partial}{\partial \mu} \left[ \frac{(1-\mu^2)}{(\eta^2 - \mu^2)} \, \frac{\partial}{\partial \mu} \right] - \frac{1}{\eta^2 - \mu^2} \, \left[ -\frac{s}{\eta} \, \frac{(\eta^2 + \mu^2)}{(\eta^2 - \mu^2)} + \frac{s^2}{1-\mu^2} \right] using \mu = \sin \varphi , \eta= \sigma / (2 \Omega) and
eigenvalue \varepsilon_n = (2 \Omega a)^2 / gh_n. Hence, atmospheric tides are eigenoscillations (
eigenmodes) of Earth's atmosphere with
eigenfunctions \Theta_n, called
Hough functions, and
eigenvalues \varepsilon_n. The latter define the
equivalent depth h_n which couples the latitudinal structure of the tides with their vertical structure.
General solution of Laplace's equation of one
solar day. Waves with positive (negative) frequencies propagate to the east (west). The horizontal dashed line is at and indicates the transition from internal to external waves. Meaning of the symbols: 'RH' Rossby-Haurwitz waves (); 'Y' Yanai waves; 'K' Kelvin waves; 'R' Rossby waves; 'DT' Diurnal tides (); 'NM' Normal modes () Longuet-Higgins has completely solved Laplace's equations and has discovered tidal modes with negative eigenvalues (Figure 2). There exist two kinds of waves: class 1 waves, (sometimes called gravity waves), labelled by positive n, and class 2 waves (sometimes called rotational waves), labelled by negative n. Class 2 waves owe their existence to the
Coriolis force and can only exist for periods greater than 12 hours (or ). Tidal waves can be either internal (travelling waves) with positive eigenvalues (or equivalent depth) which have finite vertical wavelengths and can transport wave energy upward, or external (evanescent waves) with negative eigenvalues and infinitely large vertical wavelengths meaning that their phases remain constant with altitude. These external wave modes cannot transport wave energy, and their amplitudes decrease exponentially with height outside their source regions. Even numbers of n correspond to waves symmetric with respect to the equator, and odd numbers corresponding to antisymmetric waves. The transition from internal to external waves appears at , or at the vertical wavenumber , and , respectively. of the diurnal tide (; ) (left) and of the semidiurnal tides (; ) (right) on the northern hemisphere. Solid curves: symmetric waves; dashed curves: antisymmetric waves The fundamental solar diurnal tidal mode which optimally matches the solar heat input configuration and thus is most strongly excited is the
Hough mode (1, −2) (Figure 3). It depends on
local time and travels westward with the Sun. It is an external mode of class 2 and has the eigenvalue of . Its maximum pressure amplitude on the ground is about 60 Pa.
Vertical structure equation For bounded solutions and at altitudes above the forcing region, the
vertical structure equation in its canonical form is: \frac{\partial^2 G^{\star}_n}{\partial x^2} \, + \, \alpha_n^2 \, G^{\star}_n = F_n(x) with solution G^{\star}_n (x) \sim \begin{cases} e^{-|\alpha_n| x} & \text{:} \, \alpha_n^2 0, \, \text{ propagating}\\ e^{\left( \kappa - \frac{1}{2} \right) x} & \text{:} \, h_n = H / (1- \kappa), F_n(x)=0 \, \forall x, \, \text{ Lamb waves (free solutions)} \end{cases} using the definitions \begin{align} \alpha_n^2 &= \frac{\kappa H}{h_n} - \frac{1}{4} \\ x &= \frac{z}{H} \\ G^{\star}_n &= G_n \, \varrho_o^{\frac{1}{2}} \, N^{-1} \\ F_n(x) & = - \frac{\varrho_o^{-\frac{1}{2}}}{i \sigma N} \, \frac{\partial}{\partial x} (\varrho_o J_n). \end{align}
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 h_n is linked to the vertical wavelength \lambda_{z,n}, since \alpha_n / H is the vertical wavenumber: \lambda_{z,n} = \frac{2 \pi \, H}{\alpha_n} = \frac{2 \pi \, H}{ \sqrt{\frac{\kappa H}{h_n} - \frac{1}{4}}}. For propagating solutions (\alpha_n^2 > 0), the vertical group velocity c_{gz,n}=H \frac{\partial \sigma}{\partial \alpha_n} becomes positive (upward energy propagation) only if \alpha_n > 0 for westward (\sigma or if \alpha_n for eastward (\sigma >0) propagating waves. At a given height x=z/H, the wave maximizes for K_n = s\lambda + \alpha_n x - \sigma t = 0. For a fixed longitude \lambda, 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 \propto e^{z/2H}, as density decreases. ==Dissipation==