Theoretical analysis of equatorial near-inertial solitary waves under complete Coriolis parameters

The influence of the atmosphere and ocean on human beings has always been a topic of interest for scientists (Nezlin and Snezhkin, 1993) because it is useful to understand the physical mechanisms of atmospheric or oceanic motions to further predict extreme weather or climate phenomena. Theoretically, a series of primitive partial differential equations, including the mass, momentum, state and so on, are used to characterize the motions of the atmosphere and ocean (Pedlosky, 1987). The motions of the atmosphere and ocean are affected by multiple physical factors, such as gravitation, Coriolis force, and friction. This leads to multiscale motions from turbulent microclusters and planetary waves for geophysical fluids. Existing theories have shown that it is appropriate to classify the motions of geophysics into three categories, i.e., large-scale, mesoscale and small-scale motions (Holton and Hakim, 2013). Large-scale motions, such as planetary waves, show that the rotation of the earth is extremely important; it is mainly used to disclose the physical mechanisms of the slowly varying phenomena for the atmosphere or ocean, and more details can be found in other studies (Long, 1964; Benney, 1966; Ono, 1981; Caillol and Grimshaw, 2008; Guo et al., 2019; Fu et al., 2019; Fu and Yang, 2019; Ren et al., 2019; Zhang et al., 2019a, 2019b; Zhang and Yang, 2019). Mesoscale atmospheric and oceanic motions characterize wave phenomena, such as the internal waves, under the combined actions of gravity, pressure, Coriolis force (Khater et al., 2006b; Helal and Seadawy, 2012; Seadawy, 2017a, 2018; Seadawy and Alamri, 2018; Yang et al., 2019). Mesoscale motions have received increased attention from investigators because they can help us understand the excitation, evolution, and propagation of a class of weather events, such as thunderstorms and rainstorms. Small-scale oceanic motions generally describe the fluctuations with vertical length scale from 100 m to the atomic dissipation, including the fine-structure process and micro-structure process (Caldwell, 1983).

Simplified and approximate models have always been important ways to understand various wave phenomena in the atmosphere and ocean because of the complexity of the original primitive model equations. For example, the Boussinesq approximation and shallow water approximation are usually used to model large-scale motions (Pedlosky, 1987). The shallow water approximation neglects the terms related to the horizontal component of Coriolis parameters in the momentum equations, which is often called the “traditional approximation” (Phillips, 1966). The “traditional approximation” is suitable for large-scale longwave theory, which ensures the conservation of angular momentum, vorticity and energy concerning horizontal movements without considering vertical directional motions. However, with the development of higher precision science and technology and with the requirements for more accurate weather predictions, it has been suggested that the full Coriolis force should be considered, which is called “nontraditional approximation” (White and Bromley, 1995). When the complete Coriolis parameters are considered (Gerkema et al., 2008; Leibovich and Lele, 1985), the results show that the deviation is small compared to the geostrophic wind for large-scale quasi-geostrophic motion, but it reflects the imbalance of motion, and the inclusion of complete Coriolis parameters is an important factor causing the development of weather systems. In particular, the “nontraditional approximation” is more important for near-equatorial atmosphere and ocean than that in mid-latitude, and it is noted as potential contributing to equatorial Madden-Julian Oscillation (MJO) phenomena and oceanic circulation phenomena (Hayashi and Itoh, 2012).

More importantly, the vertical velocities of mesoscale systems are larger than those of large-scale motions by one or several orders. Thus, the divergence and vorticity of mesoscale motions have larger magnitudes than those of large-scale motions. The effects of the horizontal component of Coriolis parameters on mesoscale atmospheric or oceanic motions, such as the Ekman spiral (Marshall and Schott, 1999), near-inertial waves (Zhang, 1991), deep convection (Kasahara, 2010), and internal waves (Satsuma et al., 1979; Liu et al., 2015), have received increased attention. Grimshaw (1975) studied the effect of the horizontal component of Coriolis parameters on internal gravity waves in the early stage. Fruman (2009) discussed various wave forms under the action of complete Coriolis force, such as the Kelvin wave, Rossby wave, inertial gravity wave and mixed wave. In particular, when the internal or near-inertial waves in the ocean have weak stratification, the effect of “nontraditional approximation” is more obvious compared to the case under traditional approximation. The results showed that the complete Coriolis force in the mesoscale range of motion represents a nonstatic effect, which has an important impact on the dispersion and instability of near-inertial waves (Gerkema and Shrira, 2005a, b). Yasuda and Sato (2013) discussed the effect of the horizontal component of Coriolis parameters on linear near-inertial waves. Kasahara (2003) considered the nonstatic model of mid-latitude under beta-plane approximation and illustrated the importance of the complete Coriolis force in inducing linear inertial waves (BII waves) at the boundary. White and Bromley (1995) pointed out that cumulus convection could be accompanied by adiabatic heating as hot air rises in the equatorial region through scale analysis. They noted that the horizontal component of the Coriolis parameter in the latitudinal momentum equation cannot be ignored, which is one of the reasons for the generation of near-inertial waves at the equator. Furthermore, the “nontraditional approximation” would increase inertial instability (Itano and Kasahara, 2011). Kloosterziel et al. (2007) studied the zonal symmetric inertial instability of ocean motion in the near-equatorial region through high-resolution numerical simulation. Recently, Yano (2017) considered the inertial gravity wave under the action of the complete Coriolis force. It was said that the complete Coriolis force is essentially important for the high-order problem of the degenerated system under the limit of the horizontal wavelet number. In conclusion, buoyancy and rotation are two basic factors determining mesoscale circulation, so the influences of density stratification and the horizontal component of the Coriolis parameter on mesoscale waves are quite important, especially for near-inertial waves. Thus, there are many difficulties in the study of related problems because of the density stratification of fluids and the introduction of the complete Coriolis force.

However, most existing studies neglect the nonlinear effect and vertical acceleration, which is the essence of the nonlinear wave. This paper intends to conduct a thorough study on nonlinear near-inertial waves under complete Coriolis parameters using mathematical mechanics, providing an important theoretical basis for the research and application of numerical weather forecasting and weather event phenomena in the atmosphere and ocean. The paper is organized as follows. In Section 2, a mathematical description of the atmospheric or oceanic motions is given. In Section 3, the simplified model is derived according to the scale analysis. In Section 4, a Korteweg de Vries (KdV) model equation is obtained to simulate the evolution of equatorial near-inertial waves by using the multiple scale method and perturbation expansions. The exact solutions, including the periodic and solitary solutions, for the obtained KdV equation are given. In Section 5, the dynamic effect of the horizontal component of Coriolis parameters on the equatorial near-inertial waves is assessed by numerical simulation. Conclusions are offered at the end of the paper.

In this article, we consider the mesoscale oceanic model under the complete Coriolis parameters, but without the effects of topography, dissipation, adiabatic heating or any other factors. It is written as follows in the local Cartesian coordinate system:

where x, y and z are the zonal, meridional and vertical coordinates, and u, v, w are the velocities, accordingly. ${\rho _0},{\theta _0}$ are the density and potential temperature in the ambient flow field, respectively, they are expressed as functions of $z$, and the temperature stratification is defined as $\sigma = \dfrac{{{\rm{d}}{\theta _0}}}{{{\rm{d}}z}}$. $\theta $ is the perturbation of temperature. The vertical component of the equatorial Coriolis parameters is $\,f = {f_0} + \beta y$ with $\,{f_0} = 0$, thus,

The beta parameter in the equatorial region is $\,\beta = {{2\Omega } / {{a_0}}}$, where $\Omega $ is the angular velocity of the earth’s rotation and ${a_0}$ is the radius of the earth. At the same time, the horizontal component of the Coriolis parameters is $\,f\;' \!=\! 2\Omega $, and it is constant. $p$ is the pressure, and $g$ is the gravitational acceleration.

We introduce the nondimensional quantities as follows:

where $L$ and $U$are the horizontal characteristic length and characteristic velocity, $D$ is the vertical characteristic length, $H$is the homogeneous atmospheric height, and it is assumed that $D \!=\! H$. $P$ is the characteristic pressure, and the changes of pressures in the horizontal and vertical directions are presented by $\text{δ}{p_{x,y}}$ and $\text{δ}{p_z}$. ${\rho _{\rm{s}}}$ is the nondimensional density, and $g$ is the gravitational acceleration. The quantities with apostrophes represent dimensionless ones. The nondimensional equations are obtained by substitutions of Eqs (2) and (3) into Eq. (1)

where $Ro \!= \!\dfrac{U}{{\beta {L^2}}}$ is the equatorial Rossby number, and $\gamma \! = \! \dfrac{{f\ '}}{{\beta L}}$ represents the ratio of the horizontal component to the vertical component of the equatorial Coriolis parameters; $\text{δ} \!=\! \dfrac{D}{L}$ is expressed as the aspect ratio of the motion. $\alpha \! =\! \gamma \text{δ} $ and $\lambda \!=\! \dfrac{\gamma }{\text{δ} }$ represent the combined influences of the horizontal component of Coriolis parameters and the aspect ratio on the equatorial near-inertia fluctuations. This shows that the three-dimensional fluctuation is a necessary condition for studying the complete Coriolis force, which means that it is not sufficient to add the horizontal component of the Coriolis parameters into the shallow water model in the traditional sense. Furthermore, the quantitative comparison between $\lambda $ and $\alpha $ reveals a relatively greater influence of $\lambda $ than $\alpha $ on the motion.

In fact, the condition $U\! = \!{C_{\rm{g}}}$ is almost satisfied for the equatorial atmospheres, where ${C_{\rm{g}}}$ is the characteristic velocity of the gravity wave. The time scale is

and then $T = \dfrac{L}{{{C_{\rm{g}}}}} \!= \!{(\beta {C_g})^{ - \frac{1}{2}}}$. In the lower tropical area, the following assumptions are satisfied with ${C_{\rm{g}}} \!\approx\! {10^2}\;{\rm{m}}/{{\rm{s}}},\;\beta \!\approx \!2.3 \times$${10^{ - 11}}\;({{\rm{m}}\cdot{{\rm{s}}})}^{-1},L\!\approx\! 2 \!\times \!{10^6}\;{\rm{m}},T \!\approx\! 6\;{\rm{h}},U \!\approx \!20\;{\rm{m}} /{\rm{s}}$. Thus, it is acceptable that Ro≈1 in the tropics, and ignoring Ro is not appropriate for our further study. We take $Ro\! \approx\! 1$ in the tropics, and ignoring $Ro$ is not appropriate for our further study. We take $Ro \!\approx\! 1$ in what follows according to the above discussions. The second term of the fourth equation in Eq. (4) is generally small by the scale balance principle; it is known that

thus $\dfrac{{g\text{δ} \theta }}{{\beta UD{\theta _0}}} \!=\! \dfrac{{{N^2}}}{{\beta C_{\rm{g}}^{}}}$, where ${N^2} \!=\! {{g\sigma }/{{\theta _0}}}$ is the Brunt-Väisälä frequency. Denote

then $\varepsilon \!<\! <\! 1$. When the dimensionless apostrophes are omitted, Eq. (4) becomes

Many kinds of PDEs have been derived to simulate the evolution of waves (Khater et al., 2006a; Seadawy, 2011; 2015; 2016; 2017b; Seadawy et al., 2017; Tian et al., 2019). Here, we use the KdV model equation. To balance the dispersion and nonlinearity, we introduce

where $T,\;\xi $ are slowly varying. $u,\;v,\;w,\;p,\;\theta $ are expanded:

Substitutions of Eqs (9) and (10) into Eq. (8) yield an order equation. The lowest-order problem is

Equation (11) indicates that the basic flow satisfies the equatorial semigeostrophic equilibrium and static equilibrium. When $\left| {\dfrac{1}{{\rho _{\rm{s}}^2}}\dfrac{{\partial {\rho _{\rm{s}}}}}{{\partial z}}} \right| \!< \!< \!1$, the thermal wind is

Thus, the lowest order equation reveals information of the background flow, but without any information about the perturbation flow. We further consider the higher orders with representations of

and

To address Eqs (13) and (14), we introduce

where i=0, 1. Substituting Eq. (15) into Eqs (13) and (14) converts them into the following forms:

where

and

According to Eq. (16), we have

and

Moreover,

The second equation of Eq. (16) becomes

Substitutions of Eqs (21)–(23) into the fifth equation of Eq. (16) lead to the PDEs with respect to P_i

where the operators are

For $i = 0$, it is easy to obtain

Thus, we obtain the solutions:

Further,

Thus, Eq. (25) is simplified as

where

Equation (24) has the same homogeneous term for the cases i=0 and i=1. To satisfy the nonsingularity of the solution, the following is required:

where $\tilde p_0^{\rm{*}}$ is the solution for the conjugate equation of Eq. (28). Then, Eq. (30) becomes the following form:

where the coefficients $I,\;{I_1},\;{I_2},\;{I_3}$ are

Eqation (31) is the well-known KdV equation. Special attention is paid to the coefficient I₁, which shows that the horizontal component of Coriolis parameters is also an important factor affecting near-inertial waves in the equatorial atmosphere by modifying the velocity characteristic of the waves. This is consistent with the result of Rossby waves accompanying topography under the complete Coriolis force by Yang et al. (2016). However, the topography is not a necessary factor in our present study. On the other hand, the effect of α on the wave is mainly reflected in a combined way with the vertical shear of the basic flow, i.e., ${\bar u_z}$, it is a result that cannot be obtained under the traditional approximation. When the horizontal component of the Coriolis parameters is not considered, the equations become the shallow water model, and then Eq. (31) is reduced to the traditional case.

Equation (31) is simplified into

where ${a_1} = \dfrac{{{I_2}}}{I},{a_2} = \dfrac{{{I_3}}}{I},{a_3} = \dfrac{{{I_1}}}{I}$.

Many kinds of exact analytical methods have been proposed in recent years to solve nonlinear evolution equations, such as solitary waves. In what follows, we mainly adopt the method of Jacobian elliptic function expansions to solve Eq. (32). Introduce an appropriate ansatz $X = k(\xi - {c_1}T)$, where $k$ is the wavenumber and ${c_1}$ is the wave velocity. Thus, Eq. (32) is

According to the method of Jacobian elliptic function expansions, $A(X)$ has the formal solution as a finite summation of $sn{\rm{ }}X$:

where $n \!=\! 2$ is a necessary condition for the balance between the nonlinearity and dispersion; then, we obtain

By using the properties of elliptic functions, the solution is

When $m \to 1$, the solitary wave solution is

especially, when ${c_1}\! \!- \!\!{a_3}\! -\! 4{a_2}{k^2} \!=\! 0$, the solitary solution becomes

Obviously, both the periodic and solitary wave solutions show that the effect of the Coriolis horizontal component on the equatorial nonlinear wave is mainly reflected by modifying the wave velocity but not the wave amplitude or wave energy.

In this section, we devote ourselves to the dynamic analysis of the effect of the horizontal Coriolis parameter on the nonlinear equatorial near-inertial solitary waves through numerical simulations. According to the previous scale analysis, the effect of parameter $\alpha $ is much smaller than $\lambda $. Thus, in what follows, we mainly discuss the dynamic effect of the horizontal component of Coriolis parameters through $\lambda $, which means that the coefficient of ${a_3}$ represents the main function of the Coriolis horizontal component on the evolution of equatorial near-inertial solitary waves. In addition, it is acceptable to perform some coordinate transformations so that all the coefficients of Eq. (32) become one, with the popular Fourier spectral method for Eq. (32) with fixed ${a_1} = {a_2} = 1$ and varied ${a_3}$. An initial solitary wave solution is assumed to be ${\rm{sech} ^2}\xi $.

Figure 1 depicts the whole evolution processes of the solitary equatorial near-inertial waves under the nontraditional approximation. From the figure, we can see that the initial solitary wave ${\rm{sech} ^2}\xi $ evolves to the west and east along with time, the magnitude of the amplitude decreases with time in both directions, and the solitary wave packets are excited when the solitary waves are propagating. On the other hand, it is interesting to find that the asymmetry structures of the wave packets with respect to zonal spatial coordinates are formulated during its evolutionary process. It denotes that the solitary wave is propagating to the west with higher speed and frequencies, it is consistent with some existed results, such as the work (Gerkema and Shrira, 2005a, 2005b). In order to disclose the effect of the horizontal component of the Coriolis parameters more clearly, two-dimensional projections of the propagation process are given in Fig. 2.

Figure 1.  The evolution processes of wave magnitude with time and space.

Figure 2.  The dynamic effect of the horizontal Coriolis parameter on the evolution of solitary wave.

In Fig. 2, the evolution of the solitary wave amplitude in the two-dimensional case with different ${a_3}$ is characterized, which means that the horizontal Coriolis parameter varies affects the evolution of solitary waves. We can find that the initial solitary wave evolves into wave packets to both west and east with time goes. It is obvious that ${a_3}$ affects the wave velocity. It denotes that larger magnitude of ${a_3}$ increases the westward propagation of solitary waves. Furthermore, it is to say that the dispersion relationship is essentially modified by the consideration of complete Coriolis parameters, it is consistent with the former qualitative analysis for Eq. (31).

Traditional investigations on inertial waves neglect the effect of complete Coriolis parameters, leading to qualitative inaccuracy. This paper is the first to investigate the effect of the horizontal Coriolis parameter on nonlinear equatorial near-inertial waves. Detailed scale analysis is given based on the primitive mesoscale oceanic model equations, and it denotes that a necessary condition for the inclusion of the complete Coriolis parameter is the three-dimensional oceanic motion. The evolution of the equatorial near-inertial wave was proven to be satisfied with the classical KdV model equation, and the coefficients related to the horizontal Coriolis parameter were denoted to affect the evolution of solitary waves by mainly modifying the wave speed and by interacting with the basic flow. The Fourier spectral method was used to verify the dynamic mechanisms.

In recent years, more and more investigations were finished to disclose the potential influences of complete Coriolis parameters on kinds of oceanic motions. Tort et al. (2016) and Kloosterziel et al. (2017) studied the inertial asymmetry and instability problem with complete Coriolis force by numerical simulations, the results of the present paper are consistent with those in their papers. Thus, the theoretical analysis results of this paper may be potentially contributed to the understanding of oceanic near-inertial waves. More importantly, much more attentions should be further paid to the observational or experimental results to verify the consistency of the present theoretical analysis in our further investigations.