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.^[1]

The migrating solar tides have been extensively studied both through observations and mechanistic models.^[2]

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 with longitude, land-sea contrast, and surface interactions. An important source is latent heat release due to deep convection in the tropics.

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, Africa and Australia.^[3]

Basic equations

The primitive equations lead to the linearized equations for perturbations (primed variables) in a spherical isothermal atmosphere:^[6]

with the definitions

• ${\displaystyle u}$ eastward zonal wind
• ${\displaystyle v}$ northward meridional wind
• ${\displaystyle w}$ upward vertical wind
• ${\displaystyle \Phi }$ geopotential, ${\displaystyle \int g(z,\varphi )\,dz}$
• ${\displaystyle N^{2}}$ square of Brunt-Vaisala (buoyancy) frequency
• ${\displaystyle \Omega }$ angular velocity of the Earth
• ${\displaystyle \varrho _{o}}$ density ${\displaystyle \propto \exp(-z/H)}$
• ${\displaystyle z}$ altitude
• ${\displaystyle \lambda }$ geographic longitude
• ${\displaystyle \varphi }$ geographic latitude
• ${\displaystyle J}$ heating rate per unit mass
• ${\displaystyle a}$ radius of the Earth
• ${\displaystyle g}$ gravity acceleration
• ${\displaystyle H}$ constant scale height
• ${\displaystyle t}$ time

Separation of variables

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

$${\displaystyle {\begin{aligned}\Phi '(\varphi ,\lambda ,z,t)&={\hat {\Phi }}(\varphi ,z)\,e^{i(s\lambda -\sigma t)}\\{\hat {\Phi }}(\varphi ,z)&=\sum _{n}\Theta _{n}(\varphi )\,G_{n}(z)\end{aligned}}}$$

and doing some manipulations^[7] 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:

$${\displaystyle {L}{\Theta }_{n}+\varepsilon _{n}{\Theta }_{n}=0}$$

with Laplace operator

$${\displaystyle {L}={\frac {\partial }{\partial \mu }}\left[{\frac {(1-\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}\,{\frac {\partial }{\partial \mu }}\right]-{\frac {1}{\eta ^{2}-\mu ^{2}}}\,\left[-{\frac {s}{\eta }}\,{\frac {(\eta ^{2}+\mu ^{2})}{(\eta ^{2}-\mu ^{2})}}+{\frac {s^{2}}{1-\mu ^{2}}}\right]}$$

using ${\displaystyle \mu =\sin \varphi }$, ${\displaystyle \eta =\sigma /(2\Omega )}$ and eigenvalue

$${\displaystyle \varepsilon _{n}=(2\Omega a)^{2}/gh_{n}.}$$

Hence, atmospheric tides are eigenoscillations (eigenmodes)of Earth's atmosphere with eigenfunctions ${\displaystyle \Theta _{n}}$, called Hough functions, and eigenvalues ${\displaystyle \varepsilon _{n}}$. The latter define the equivalent depth ${\displaystyle h_{n}}$ which couples the latitudinal structure of the tides with their vertical structure.

General solution of Laplace's equation

Longuet-Higgins^[8] has completely solved Laplace's equations and has discovered tidal modes with negative eigenvalues ε s
n  (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 |ν| ≤ 2). 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 ε ≃ ε_c, or at the vertical wavenumber k_z = 0, and λ_z ⇒ ∞, respectively.

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 ε 1
−2  = −12.56. Its maximum pressure amplitude on the ground is about 60 Pa.^[5] The largest solar semidiurnal wave is mode (2, 2) with maximum pressure amplitudes at the ground of 120 Pa. It is an internal class 1 wave. Its amplitude increases exponentially with altitude. Although its solar excitation is half of that of mode (1, −2), its amplitude on the ground is larger by a factor of two. This indicates the effect of suppression of external waves, in this case by a factor of four.^[9]

Vertical structure equation

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

$${\displaystyle {\frac {\partial ^{2}G_{n}^{\star }}{\partial x^{2}}}\,+\,\alpha _{n}^{2}\,G_{n}^{\star }=F_{n}(x)}$$

with solution

$${\displaystyle G_{n}^{\star }(x)\sim {\begin{cases}e^{-|\alpha _{n}|x}&{\text{:}}\,\alpha _{n}^{2}<0,\,{\text{ evanescent or trapped}}\\e^{i\alpha _{n}x}&{\text{:}}\,\alpha _{n}^{2}>0,\,{\text{ propagating}}\\e^{\left(\kappa -{\frac {1}{2}}\right)x}&{\text{:}}\,h_{n}=H/(1-\kappa ),F_{n}(x)=0\,\forall x,\,{\text{ Lamb waves (free solutions)}}\end{cases}}}$$

using the definitions

$${\displaystyle {\begin{aligned}\alpha _{n}^{2}&={\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}\\x&={\frac {z}{H}}\\G_{n}^{\star }&=G_{n}\,\varrho _{o}^{\frac {1}{2}}\,N^{-1}\\F_{n}(x)&=-{\frac {\varrho _{o}^{-{\frac {1}{2}}}}{i\sigma N}}\,{\frac {\partial }{\partial x}}(\varrho _{o}J_{n}).\end{aligned}}}$$

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 ${\displaystyle h_{n}}$ is linked to the vertical wavelength ${\displaystyle \lambda _{z,n}}$, since ${\displaystyle \alpha _{n}/H}$ is the vertical wavenumber:

$${\displaystyle \lambda _{z,n}={\frac {2\pi \,H}{\alpha _{n}}}={\frac {2\pi \,H}{\sqrt {{\frac {\kappa H}{h_{n}}}-{\frac {1}{4}}}}}.}$$

For propagating solutions ${\displaystyle (\alpha _{n}^{2}>0)}$, the vertical group velocity

$${\displaystyle c_{gz,n}=H{\frac {\partial \sigma }{\partial \alpha _{n}}}}$$

becomes positive (upward energy propagation) only if ${\displaystyle \alpha _{n}>0}$ for westward ${\displaystyle (\sigma <0)}$ or if ${\displaystyle \alpha _{n}<0}$ for eastward ${\displaystyle (\sigma >0)}$ propagating waves. At a given height ${\displaystyle x=z/H}$, the wave maximizes for

$${\displaystyle K_{n}=s\lambda +\alpha _{n}x-\sigma t=0.}$$

For a fixed longitude ${\displaystyle \lambda }$, 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 ${\displaystyle \propto e^{z/2H}}$, as density decreases.