1 Introduction

Jerks and hyperjerk are physical quantities having a particular importance in sciences and engineering since they arise in dynamical systems, practical engineering and transport engineering characterized by rotating or sliding pieces, e.g. robotics, planning of tracks and roads (Sparavigna 2015). They also play a significant role in plasma medium (Wharton et al. 2013, 2014), in electrodynamics (Xu et al. 2009) and in chaotic dynamics (Chlouverakis and Sprott 2006; Linz 1997; Sprott 1997; Vaidyanathan et al. 2018). One of the most important manifestations of jerks is “geomagnetic jerks” or “geomagnetic impulses” which are unexpected changes in the second time-derivative (secular acceleration) of the Earth’s magnetic field due to a change in the fluid flow at the surface of the Earth’s core driven by both thermal and compositional gradients (Malin and Hodder 1982; Bloxham et al. 2002; Malkin 2013). The Earth’s magnetic field changes on long and short timescales and it’s the secular acceleration which is used to study the variations of the magnetic field over time (Cox and Brown 2013). Due to this variation in time, the biological and tectonic evolutions during geological times occur (Meyerhoff et al. 1996). This is an important problem since the correlation between all these features allows researchers to predict a number of future events occurring on our Earth, e.g. the present global warming due to the Earth’s magnetic field decrease (Rousseau 2005). In fact, most of the approaches dealing with jerks and hyperjerk in rotational dynamics are based on numerical simulations and computational analysis and there is an absence of a global mathematical theory modeling the Earth’s dynamics. Undoubtedly, such a mathematical theory is characterized by a large number of parameters and its construction is somewhat tricky. In the present work, we will try to construct an initial theory based on non-local-in-time kinetic energy approach introduced recently by Suykens where jerks and hyperjerk arise naturally from nonlocality (Suykens 2009).

In fact, nonlocal effects occur in a large number of physical phenomena, e.g. turbulent channel flows (Speziale and Eringen 1981), nonisothermal hydrodynamics (Hutter and Brader 2009), gyrokinetic turbulence (Gorler et al. 2010), fluid dynamics (Morozov 1983; Martinez 2010), hydrodynamics of nonrelativistic plasma (Dinariev 1996), among others (Cemal Eringen 1972; Rudyak and Yanenko 1985). Due to their importance in physical sciences, we will construct in this paper a nonlocal rotational dynamics model based on Suykens’s approach where nonlocality-in-time emerge effortlessly (Suykens 2009) and we will discuss its relevance on Earth’s dynamics. In fact, Suykens’s approach is based in fact on the notion of backward–forward shifting coordinates in such a way the kinetic energy \(\tfrac{1}{2}mvv\) is substituted by \(\tfrac{1}{2}mv\tfrac{v(t + \tau ) + v(t - \tau )}{2}\) where \(\tau\) is a time parameter considered to be small relative to the time scale of the dynamical system. The physical motivation behind the switch from \(v(t)\) to \(\tfrac{v(t + \tau ) + v(t - \tau )}{2}\) is motivated from Feynman’s observation of the kinetic energy term which can be written as a discrete time numerical estimation to velocities \(\tfrac{1}{2}m\tfrac{{r_{k + 1} - r_{k} }}{\varepsilon }\tfrac{{r_{k} - r_{k - 1} }}{\varepsilon }\) with \(\varepsilon = \Delta t_{i} = t_{i + 1} - t_{i}\) in association to a measurement quantum process by means of the kinetic energy of the particle. The interpretation of the parameter \(\tau\) depends on the application but in all cases its presence leads to a deviation from the standard approach. This replacement is entitled “nonlocal-in-time kinetic energy approach” since after performing Taylor series expansions of \(v(t \pm \tau ) = \dot{x}(t \pm \tau )\) where mathematically \(x(t + \tau ) = x(t) + \sum\nolimits_{k = 1}^{n} {{{\tau^{n} x^{(n)} (t)} \mathord{\left/ {\vphantom {{\tau^{n} x^{(n)} (t)} {n!}}} \right. \kern-0pt} {n!}}}\) and \(x(t - \tau ) = x(t) + \sum\nolimits_{k = 1}^{n} {{{( - 1)^{n} \tau^{n} x^{(n)} (t)} \mathord{\left/ {\vphantom {{( - 1)^{n} \tau^{n} x^{(n)} (t)} {n!}}} \right. \kern-0pt} {n!}}}\), the factor \(\tfrac{v(t + \tau ) + v(t - \tau )}{2} = v + \tfrac{{\tau^{2} }}{2}J + {\text{O}}(\tau^{3} )\) where \(J \equiv x^{(3)}\) is the jerk. The kinetic energy becomes nonlocal-in-time and takes the form \(T_{\tau ,n} = \tfrac{1}{2}mv^{2} + \tfrac{1}{4}mv\sum\nolimits_{k = 1}^{n} {(1 + ( - 1)^{k} )\tfrac{{\tau^{k} }}{k!}x^{(k + 1)} }\). The occurrence of a jerk term (a higher-order derivative term) is interesting since it leads to a number of motivating features not found in the standard formalism, e.g. discretization of the classical system, quantization of the nonlocality-in-time extent, materialization of a quantum acceleratum operator among others (Li et al. 2009; El-Nabulsi 2015, 2017a; b, c, d; El-Nabulsi 2018a, b, c, d, e, f, g; Kamalov 2013). It is noteworthy that for \(n > 2\), hyperjerk occurs as well, e.g. snap, crackle and pop in Suykens’s formalism. Since jerk and hyperjerk are important in geodynamics and in the context of rotational systems, it is clear that the implications of Suykens’s approach in these dynamics are particularly advisable. Therefore we are motivated to study in a rigorous and self-contained way the implications of nonlocal-in-time kinetic energy approach in rotational dynamics globally and to explore their impacts on Earth’s rotation. The rotating frame is expressed in terms of the Cartesian coordinate \(\vec{r} = (x,y,z)\) and the Lagrangian for a particle of mass \(m\) moving in a non-inertial rotating frame with angular velocity vector \(\vec{\omega }\) and with its origin coinciding with the fixed-frame origin in the presence of the potential \(V(\vec{r})\) is expressed generally as (Brizard 2004):

$$\begin{aligned} L(\vec{r},\dot{\vec{r}}) & = \frac{m}{2}\left| {\dot{\vec{r}} + \vec{\omega } \times \vec{r}} \right|^{2} - V(\vec{r}), \\ & = \frac{m}{2}\left( {\left| {\dot{\vec{r}}} \right|^{2} + 2\vec{\omega } \cdot \left( {\vec{r} \times \dot{\vec{r}}} \right) + \omega^{2} r^{2} - \left( {\vec{\omega } \cdot \vec{r}} \right)^{2} } \right) - V(\vec{r}). \\ \end{aligned}$$
(1)

The transition from local to nonlocal-in-time Lagrangian is done after simply replacing \(\tfrac{1}{2}m\left| {\dot{\vec{r}}^{2} } \right|\) by \(\tfrac{1}{2}m\left| {\dot{\vec{r}}^{2} } \right| + \tfrac{1}{4}m\vec{r}\sum\nolimits_{k = 1}^{n} {(1 + ( - 1)^{k} )\tfrac{{\tau^{k} }}{k!}\vec{r}^{(k + 1)} }\) and therefore Eq. (1) takes now the following form:

$$L_{\tau ,n} (\vec{r},\dot{\vec{r}}) = \frac{m}{2}\left( {\left| {\dot{\vec{r}}} \right|^{2} + \frac{1}{2}\vec{r}\sum\limits_{k = 1}^{n} {(1 + ( - 1)^{k} )} \frac{{\tau^{k} }}{k!}\vec{r}^{(k + 1)} + 2\vec{\omega } \cdot \left( {\vec{r} \times \dot{\vec{r}}} \right) + \omega^{2} r^{2} - \left( {\vec{\omega } \cdot \vec{r}} \right)^{2} } \right) - V(\vec{r}).$$
(2)

Obviously, for \(n = 1\) we find \(L_{\tau ,1} (\vec{r},\dot{\vec{r}}) = L(\vec{r},\dot{\vec{r}})\) whereas for \(n = 2\) we obtain:

$$L_{\tau ,2} (\vec{r},\dot{\vec{r}}) = \frac{m}{2}\left( {\left| {\dot{\vec{r}}} \right|^{2} + \frac{1}{2}\tau^{2} \dot{\vec{r}} \cdot \vec{J}_{{\vec{r}}} + 2\vec{\omega } \cdot \left( {\vec{r} \times \dot{\vec{r}}} \right) + \omega^{2} r^{2} - \left( {\vec{\omega } \cdot \vec{r}} \right)^{2} } \right) - V(\vec{r}),$$
(3)

where \(\vec{J}_{{\vec{r}}} = \vec{r}^{(3)}\) is the rotational jerk. Since the Lagrangian contains higher-order derivative terms, the derivation of the equation of motion is done by means of the higher-order Euler–Lagrange equation:

$$\sum\limits_{j = 0}^{n + 1} {\left( { - 1} \right)^{j} \frac{{d^{j} }}{{dt^{j} }}\frac{{\partial L_{\tau ,n} }}{{\partial \vec{r}^{(j)} }}} = 0.$$
(4)

Through this work, we limit our analysis up to \(n = 2\) since for \(n > 2\), terms \(\tau^{3}\), \(\tau^{4} , \ldots\) may be considered negligible. Therefore we find effortlessly:

$$\frac{m}{4}\tau^{2} \vec{r}^{(4)} + m\ddot{\vec{r}} = - \nabla V(\vec{r}) - m\left( {\dot{\vec{\omega }} \times \vec{r} + 2\vec{\omega } \times \dot{\vec{r}} + \vec{\omega } \times \vec{\omega } \times \vec{r}} \right).$$
(5)

This equation holds 4th-order derivative term (snap) as it is expected since in general, any system described by a 3rd-order Lagrangian leads to a 4th-order differential equation. The impacts of the “snap differential equation” (SDE) (5) on rotational dynamics will be considered in this work. Equation (5) belongs to the class of 4th-order differential equation and can be regarded as 4th-order extension of Newton’s law. We will show that Eq. (5) holds important properties which the 2nd-order standard equation in general does not possess. It is noteworthy that 4th-order differential equations provide the possibility of chaotic behaviour whereas 2nd-order equations are unable to supply chaotic behaviour since the system does not hold an adequate amount of degrees of freedom. The diversity of models that evolve in the 4th-order differential equations is much larger than in the 2nd-order differential equations (van den Berg 2000). Therefore it will be of interest to explore their impacts on rotational dynamics. Besides, the rotation of the Earth represents an interesting fluid flow problem in fluid dynamics. Since in most applied fluid dynamics phenomena, e.g. oceanography and meteorology, the frame of reference is not inertial, it will be motivating to examine how the nonlocal equations of motion must be altered to take this into account and to study the behaviors of fluids in rotating reference systems. In other words, we will construct the nonlocal Navier–Stokes equations of fluid dynamics under the influence of Earth rotation and will study their properties in the rotating frame.

The paper is organized as follows: in Sect. 2, the nonlocal motion relative to Earth is considered and a number of illustrations were discussed based on the nonlocal-in-time approach in particular the free-fall problem, the Foucault pendulum dynamics and the motion of a particle in a rotating tube; in Sect. 3, we construct the governing fluid dynamical equations, in particular the nonlocal-in-time Navier–Stokes equations under the influence of Earth rotation, and we analyze their properties and features; conclusions and perspectives are given in Sect. 4.

2 Nonlocal Motion Relative to Earth

In order to apply Suykens’s formalism to describe the nonlocal motion relative to Earth (having mass \(M\)), we consider a fixed frame of reference having its origin at the center of Earth and the rotating frame of reference having its origin at latitude \(\lambda\) and longitude \(\psi\) such that the angular velocity of the Earth is \(\omega = \dot{\psi }\) where \(\ddot{\psi } = 0\). The rotating frame of the Earth is \(R(x^{\prime},y^{\prime},z^{\prime})\), the radius of the Earth is \(R\), the position vector is in the \(z\)-direction such that \(\vec{R} = R\hat{z}\) and the \(x\)-axis is tangent to a great circle passing through the North and South poles. The transition matrix from \(R(x,y,z) \to R(x^{\prime},y^{\prime},z^{\prime})\) is (Brizard 2004):

$$\left( {\begin{array}{*{20}c} {\hat{x}} \\ {\hat{y}} \\ {\hat{z}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {\sin \lambda \cos \psi } & {\sin \lambda \sin \psi } & { - \cos \lambda } \\ { - \sin \psi } & {\cos \psi } & 0 \\ {\cos \lambda \cos \psi } & {\cos \lambda \sin \psi } & {\sin \lambda } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {\hat{x}^{\prime}} \\ {\hat{y}^{\prime}} \\ {\hat{z}^{\prime}} \\ \end{array} } \right),$$
(6)

The SDE of a point P as observed in the rotating frame is given by:

$$\frac{1}{4}\tau^{2} \vec{r}^{(4)} + \ddot{\vec{r}} = - \frac{GM}{{\left| {\vec{r}^{\prime}} \right|^{3} }}\vec{r}^{\prime} - \ddot{\vec{R}}_{f} - 2\vec{\omega } \times \dot{\vec{r}} - \vec{\omega } \times \vec{\omega } \times \vec{r},$$
(7)

where \(\vec{r}^{\prime} = \vec{R} + \vec{r}\) is the position of point P in the fixed frame, \(\vec{r}\) is its location in the rotating frame and \(G\) is the universal gravitational constant. The 1st term on the RHS of Eq. (7) is the gravitational acceleration due to the gravitational drag of the Earth on point P as observed in the fixed frame located at the center of the Earth. It is expressed in spherical coordinates \((r,\theta ,\varphi )\) by:

$$\vec{g}_{0} = - \frac{GM}{{R^{2} }}\frac{{\left( {1 + \varepsilon \cos \theta } \right)\hat{z} + \varepsilon \sin \theta \left( {\cos \varphi \hat{x} + \sin \varphi \hat{y}} \right)}}{{\left( {1 + 2\varepsilon \cos \theta + \varepsilon^{2} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} }},$$
(8)

where \(\varepsilon = {r \mathord{\left/ {\vphantom {r R}} \right. \kern-0pt} R}\) is a tiny parameter. In the rotating frame, we have in addition \(\vec{\omega } = \omega \hat{z}^{\prime} = \omega (\sin \lambda \hat{z} - \cos \lambda \hat{x})\),\(\omega = 7.27 \times 10^{ - 5} {\text{ rad s}}^{ - 1}\) being the Earth’s angular velocity about its axis. Since \(\vec{R}\) rotates with the origin of the rotating frame, it is easy to check that \(\dot{\vec{R}}_{f} = \vec{\omega } \times \vec{R}\) and \(\ddot{\vec{R}}_{f} = \vec{\omega } \times \dot{\vec{R}} = R\omega^{2} \cos \lambda (\cos \lambda \hat{z} + \sin \lambda \hat{x})\). Here \(R\omega^{2} = 3.4 \times 10^{ - 3} g_{0}\) where \(g_{0} = {{GM} \mathord{\left/ {\vphantom {{GM} {R^{2} }}} \right. \kern-0pt} {R^{2} }} \approx 9.8{\text{ m s}}^{ - 2}\). To the lowest order in \(\varepsilon\), we can write Eq. (7) as:

$$\frac{1}{4}\tau^{2} \vec{r}^{(4)} + \ddot{\vec{r}} = - g\hat{z} - 2\omega \left( { - \dot{y}\sin \lambda \hat{x} + \left( {\dot{x}\sin \lambda + \dot{z}\cos \lambda } \right)\hat{y} - \dot{y}\cos \lambda \hat{z}} \right),$$
(9)

which gives respectively the following components:

$$\frac{1}{4}\tau^{2} x^{(4)} + \ddot{x} = 2\omega \dot{y}\sin \lambda ,$$
(10)
$$\frac{1}{4}\tau^{2} y^{(4)} + \ddot{y} = - 2\omega \left( {\dot{x}\sin \lambda + \dot{z}\cos \lambda } \right),$$
(11)
$$\frac{1}{4}\tau^{2} z^{(4)} + \ddot{z} = - g + 2\omega \dot{y}\cos \lambda .$$
(12)

The integration of Eqs. (10)–(12) gives:

$$\frac{1}{4}\tau^{2} \ddot{x} + x = x_{0} + k_{x} t + 2\omega \sin \lambda \int\limits_{0}^{t} {ydt} \equiv x_{0} + k_{x} t + \delta x,$$
(13)
$$\frac{1}{4}\tau^{2} \ddot{y} + y = y_{0} + k_{y} t - 2\omega \sin \lambda \int\limits_{0}^{t} {xdt} - 2\omega \cos \lambda \int\limits_{0}^{t} {zdt} \equiv y_{0} + k_{y} t + \delta y,$$
(14)
$$\frac{1}{4}\tau^{2} \ddot{z} + z = z_{0} + k_{z} t - \frac{1}{2}gt^{2} + 2\omega \cos \lambda \int\limits_{0}^{t} {ydt} \equiv z_{0} + k_{z} t - \frac{1}{2}gt^{2} + \delta z.$$
(15)

Here \(k_{x}\), \(k_{y}\) and \(k_{z}\) are constants of integrations such that:

$$k_{x} = \frac{1}{4}\tau^{2} x_{0}^{(3)} + \dot{x}_{0} - 2\omega y_{0} \sin \lambda ,$$
(16)
$$k_{y} = \frac{1}{4}\tau^{2} y_{0}^{(3)} + \dot{y}_{0} + 2\omega \left( {x_{0} \sin \lambda + z_{0} \cos \lambda } \right),$$
(17)
$$k_{z} = \frac{1}{4}\tau^{2} z_{0}^{(3)} + \dot{z}_{0} - 2\omega y_{0} \cos \lambda .$$
(18)

\(\delta x\), \(\delta y\) and \(\delta z\) are in their turns such that:

$$\delta x = 2\omega \sin \lambda \int\limits_{0}^{t} {ydt} = 2\omega \sin \lambda \left( {y_{0} t + \frac{1}{2}k_{y} t^{2} + \int\limits_{0}^{t} {\delta ydt} - \frac{1}{4}\tau^{2} \left( {\dot{y} - \dot{y}_{0} } \right)} \right),$$
(19)
$$\begin{aligned} \delta y & = - 2\omega \sin \lambda \int\limits_{0}^{t} {xdt} - 2\omega \cos \lambda \int\limits_{0}^{t} {zdt} , \\ & = - 2\omega \sin \lambda \left( {x_{0} t + \frac{1}{2}k_{x} t^{2} + \int\limits_{0}^{t} {\delta xdt} - \frac{1}{4}\tau^{2} \left( {\dot{x} - \dot{x}_{0} } \right)} \right) \\ & \quad - 2\omega \cos \lambda \left( {z_{0} t + \frac{1}{2}k_{z} t^{2} - \frac{1}{6}gt^{3} + \int\limits_{0}^{t} {\delta zdt} - \frac{1}{4}\tau^{2} \left( {\dot{z} - \dot{z}_{0} } \right)} \right), \\ \end{aligned}$$
(20)

and

$$\delta z = 2\omega \cos \lambda \int\limits_{0}^{t} {ydt} = 2\omega \cos \lambda \left( {y_{0} t + \frac{1}{2}k_{y} t^{2} + \int\limits_{0}^{t} {\delta ydt} - \frac{1}{4}\tau^{2} \left( {\dot{y} - \dot{y}_{0} } \right)} \right).$$
(21)

In what follows we discuss a number of familiar problems:

2.1

As a 1st illustration, we consider the free-fall problem assuming the initial conditions \((x_{0} ,y_{0} ,z_{0} ) = (0,0,h)\) and \((\dot{x}_{0} ,\dot{y}_{0} ,\dot{z}_{0} ) = (\ddot{x}_{0} ,\ddot{y}_{0} ,\ddot{z}_{0} ) = (\dddot x_{0} ,\dddot y_{0} ,\dddot z_{0} ) = (0,0,0)\). This consequently yields \(k_{x} = k_{z} = 0\) and \(k_{y} = 2\omega h\cos \lambda\). We find up to 1st-order in \(\omega\) and neglecting terms on \(\tau^{2} \omega\):

$$\frac{1}{4}\tau^{2} \ddot{x} + x = 0,$$
(22)
$$\frac{1}{4}\tau^{2} \ddot{y} + y = \frac{1}{3}\left( {g\omega \cos \lambda } \right)t^{3} ,$$
(23)
$$\frac{1}{4}\tau^{2} \ddot{z} + z = h - \frac{1}{2}gt^{2} .$$
(24)

The approximate solutions of Eqs. (22)–(24) are respectively given by:

$$x(t) = 0,$$
(25)
$$y(t) = \frac{1}{3}\left( {g\omega \cos \lambda } \right)t^{3} ,$$
(26)
$$z(t) = h - \frac{1}{2}gt^{2} + \frac{1}{2}g\tau^{2} \sin^{2} \left( {\frac{t}{\tau }} \right).$$
(27)

The body touches the ground after a certain time \(T\) which satisfies the relation:

$$h - \frac{1}{2}gT^{2} + \frac{1}{2}g\tau^{2} \sin^{2} \left( {\frac{T}{\tau }} \right) = 0.$$
(28)

Letting \(h = 100\,{\text{m}}\) and \(\tau = 10^{ - 1} \,{\text{s}}\), we find \(T \approx 4.52106{\kern 1pt} {\text{s}}\) and therefore for \(\lambda = 45^{ \circ }\) we find an eastward drift of about \(1. 5 5\,{\text{cm}}\) whereas for \(\tau = 10^{ - 2} s\), we find \(T \approx 4.52008\,{\text{s}}\) and an eastward drift of about \(1. 5 4\,{\text{cm}}\) and finally for \(\tau = 1\,{\text{s}}\), we get \(T \approx 4.62862\,{\text{s}}\) and an eastward drift of about \(1. 6 6\,{\text{cm}}\). In the standard approach, i.e. \(\tau = 0\,{\text{s}}\), we find an eastward drift of about \(1.55\,{\text{cm}}\). Therefore the dynamics is therefore slightly affected by nonlocality.

Remark 2.1

Currently, the value of the nonlocal-in-time parameter \(\tau\) is choosen. Since in the standard approach, the eastward drift is not exact (measurement error since variability is an natural part of the results of measurements), Eq. (28) may be used to estimate the error in the measurement and consequently the parameter \(\tau\) from experimental data.

We first plot in Figs. 1 and 2 the variations of \(z(t)\) for \(\tau = 0\) (standard result), \(\tau = 0.1\,{\text{s}}\) and \(\tau = 0.01\,{\text{s}}\) for different ranges of time.

Fig. 1
figure 1

Variations of \(z(t)\) for \(\tau = 0\)(blue graph), \(\tau = 0.1\,{\text{s}}\)(yellow graph) and \(\tau = 0.01\,{\text{s}}\)(red line) for small range of time

Fig. 2
figure 2

Variations of \(z(t)\) for \(\tau = 0\)(blue graph), \(\tau = 0.1\,{\text{s}}\)(yellow graph) and \(\tau = 0.1\,{\text{s}}\)(red line)

We plot in Figs. 3, 4, 5, 6, 7, 8, 9, 10 the variations in 3D of \(z(t)\) for different ranges of time.

Fig. 3
figure 3

Variations of \(z(t)\) for \(0 < t < 1\) and \(0 < \tau < 1\)

Fig. 4
figure 4

Contour plot of Fig. 3

Fig. 5
figure 5

Variations of \(z(t)\) for \(0 < t < 1\) and \(0 < \tau < 0.01\)

Fig. 6
figure 6

Contour plot of Fig. 5

Fig. 7
figure 7

Variations of \(z(t)\) for \(0 < t < 0.005\) and \(0 < \tau < 0.005\)

Fig. 8
figure 8

Contour plot of Fig. 7

Fig. 9
figure 9

General variations of \(z(t)\)

Fig. 10
figure 10

Contour plot of Fig. 9

We observe that for \(t < < 1\), the dynamics deviates from the standard approach yet for large time the dynamics come close to the standard result. Nonlocality-in-time affects the dynamics merely for large time. Besides, for the dynamics is perturbed and there is a presence of a disorder motion. Moreover, we observe from Fig. 2 that for a small range of preliminary time, the body exhibits a small leap before falling down and touching the Earth. This weird behaviour is due to the nonlocality-in-time aspect. For large time and small \(\tau\), the dynamics is comparatively similar to the standard free-fall problem. These plots prove the sensitivity of the trajectory of the free-falling body for initial conditions and nonlocality. To clarify this point, we assume the initial conditions \((x_{0} ,y_{0} ,z_{0} ) = (0,0,h)\), \((\dot{x}_{0} ,\dot{y}_{0} ,\dot{z}_{0} ) = (0,1,0)\) and \((\ddot{x}_{0} ,\ddot{y}_{0} ,\ddot{z}_{0} ) = (\dddot x_{0} ,\dddot y_{0} ,\dddot z_{0} ) = (0,0,0)\). This yields \(k_{x} = k_{z} = 0\), \(k_{y} = 1 + 2\omega h\cos \lambda\) and consequently we obtain:

$$\frac{1}{4}\tau^{2} \ddot{x} + x = \omega \sin \lambda t^{2} ,$$
(29)
$$\frac{1}{4}\tau^{2} \ddot{y} + y = t + \frac{1}{3}\omega gt^{3} \cos \lambda ,$$
(30)
$$\frac{1}{4}\tau^{2} \ddot{z} + z = h - \frac{1}{2}gt^{2} + \omega t^{2} \cos \lambda .$$
(31)

Up to 1st-order in \(\omega\) and neglecting terms on \(\tau^{2} \omega\), the respective solutions are approximately given by:

$$x(t) = \omega \sin \lambda t^{2} ,$$
(32)
$$y(t) = t + \frac{1}{3}\left( {g\omega \cos \lambda } \right)t^{3} ,$$
(33)
$$z(t) = \left( {\omega \cos \lambda - \frac{1}{2}g} \right)t^{2} + h + \frac{1}{2}g\tau^{2} \sin^{2} \left( {\frac{t}{\tau }} \right).$$
(34)

If \(h = 100\,{\text{m}}\) and \(\tau = 10^{ - 1} \,{\text{s}}\), the equation \(z(T) = 0\) gives \(T \approx 4.51854\,{\text{s}}\) and therefore for \(\lambda = 45^{ \circ }\) we find an eastward drift of about \(6.06 \, cm\) whereas for \(\tau = 0.5\,{\text{s}}\), we find \(T \approx 4.52142\,{\text{s}}\) and an eastward drift of about \(6. 1 2\,{\text{cm}}\). The dynamics is slightly affected by nonlocality and initial conditions. We plot in Figs. 11 and 12 the variations of \(z(t)\) for \(\tau = 0\)(standard result), \(\tau = 0.1\,{\text{s}}\) and \(\tau = 0.01\,{\text{s}}\) for different ranges of time.

Fig. 11
figure 11

Variations of \(z(t)\) for \(\tau = 0\)(blue graph), \(\tau = 0.1\,{\text{s}}\)(yellow graph) and \(\tau = 0.1\,{\text{s}}\)(red line) for small range of time

Fig. 12
figure 12

Variations of \(z(t)\) for \(\tau = 0\)(blue graph), \(\tau = 0.1\,{\text{s}}\)(yellow graph) and \(\tau = 0.1\,{\text{s}}\)(red line)

The variations of \(z(t)\) in 3D for different ranges of time are similar to those illustrated in Figs. 3, 4, 5, 6, 7, 8, 9, 10, yet the general variations differ slightly as illustrated in Figs. 13 and 14.

Fig. 13
figure 13

General variations of \(z(t)\)

Fig. 14
figure 14

Contour plot of Fig. 13

There exist an additional observation which concerns the variations of \(y = f(x)\) and \(z = f(x)\). Equations (32)–(34) give:

$$y(t) = \sqrt {\frac{x}{\omega \sin \lambda }} + \frac{1}{3}\left( {g\omega \cos \lambda } \right)\left( {\frac{x}{\omega \sin \lambda }} \right)^{{\tfrac{3}{2}}} ,$$
(35)
$$z(t) = \left( {\omega \cos \lambda - \frac{1}{2}g} \right)\frac{x}{\omega \sin \lambda } + h + \frac{1}{2}g\tau^{2} \sin^{2} \left( {\frac{1}{\tau }\sqrt {\frac{x}{\omega \sin \lambda }} } \right).$$
(36)

Such forms of relations are merely absent if we choose \((x_{0} ,y_{0} ,z_{0} ) = (0,0,h)\) and \((\dot{x}_{0} ,\dot{y}_{0} ,\dot{z}_{0} ) = (\ddot{x}_{0} ,\ddot{y}_{0} ,\ddot{z}_{0} ) = (\dddot x_{0} ,\dddot y_{0} ,\dddot z_{0} ) = (0,0,0)\). We observe that \(y = f(x)\) is not effected by nonlocality in contrast to \(z = f(x)\). The variation of \(y = f(x)\) is plotted in Fig. 15 whereas the variations of \(z = f(x)\) are plotted in Figs. 16 and 17 for \(0 < x < 1\) and \(0 < \tau < 1\).

Fig. 15
figure 15

Variations of \(y = f(x)\)

Fig. 16
figure 16

Variations of \(z = f(x)\)

Fig. 17
figure 17

Contour plot of Fig. 16

The following statement holds consequently:

Statement 1

The nonlocality-in-time can be quantified in rotational dynamics and its effects are computable at small ranges of time and are very sensitive for initial conditions. The distance Eastwards from the bottom of the tower of height at which the particle lands \(h = 100\,{\text{m}}\) is about \(1. 6 6\,{\text{cm}}\) for a nonlocal-in-time parameter \(\tau = 1\,{\text{s}}\) and therefore the nonlocality effect is about \(7\%\) whereas for \(\tau = 10^{ - 2} \,{\text{s}}\) the nonlocality effect represents about \(6.5\%\). The rotational dynamics is therefore slightly affected by nonlocality and initial conditions.

2.2

As a 2nd-illustration, it is worthy to evaluate the effect of nonlocality-in-time on the dynamics of the Foucault pendulum. The corresponding nonlocal equation of motion is given by:

$$\frac{1}{4}\tau^{2} \vec{r}^{(4)} + \ddot{\vec{r}} = \vec{g} + \frac{{\vec{T}}}{m} - 2\vec{\omega } \times \dot{\vec{r}},$$
(37)

where \(m\) is the mass of the pendulum, \(\vec{T}\) is the string tension and \(\vec{g}\) is the gravitational acceleration. In the absence of nonlocality-in-time, i.e. \(\tau = 0\), it is well-known that the precession angular frequency of the Foucault pendulum is given by \(\dot{\varphi } = \omega \sin \lambda\)(Brizard 2004). In order to verify the effect of nonlocality-in-time on this result, we use the Cartesian coordinates and we split Eq. (37) in the \((x,y)\) plane into two differential equations:

$$\frac{1}{4}\tau^{2} x^{(4)} + \ddot{x} + \omega_{0}^{2} x = 2\omega \sin \lambda \dot{y},$$
(38)
$$\frac{1}{4}\tau^{2} y^{(4)} + \ddot{y} + \omega_{0}^{2} y = - 2\omega \sin \lambda \dot{x},$$
(39)

where \(\omega_{0}^{2} = {T \mathord{\left/ {\vphantom {T {ml}}} \right. \kern-0pt} {ml}} \approx {g \mathord{\left/ {\vphantom {g l}} \right. \kern-0pt} l}\) and \(\dot{z} \approx 0\) for \(l > > 1\). By letting \(Q = y + ix = l\sin \theta e^{i\varphi }\) we can write Eqs. (38) and (39) as:

$$\frac{1}{4}\tau^{2} Q^{(4)} (t) + \ddot{Q}(t) - 2i\omega \sin \lambda \dot{Q}(t) + \omega_{0}^{2} Q(t) = 0.$$
(40)

This is the equation of a damped Pais–Uhlenbeck oscillator which is a prototype of higher-order differential equation (Nesterenko 2007; Pavsic 2013). We can use this equation to evaluate the nonlocal precession angular frequency of the Foucault pendulum. The solution is given by:

$$\begin{aligned} Q(t) & = c_{1} e^{{ - \tfrac{1}{2}\left( {\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} + \sqrt { - \tfrac{4ib}{{a\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} }} - \tfrac{B}{C} - \tfrac{A}{D} - \tfrac{4}{3a}} } \right)t}} + c_{2} e^{{\tfrac{1}{2}\left( { - \sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} + \sqrt { - \tfrac{4ib}{{a\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} }} - \tfrac{B}{C} - \tfrac{A}{D} - \tfrac{4}{3a}} } \right)t}} \\ & \quad + c_{3} e^{{\tfrac{1}{2}\left( {\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} - \sqrt { - \tfrac{4ib}{{a\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} }} - \tfrac{B}{C} - \tfrac{A}{D} - \tfrac{4}{3a}} } \right)t}} + c_{4} e^{{\tfrac{1}{2}\left( {\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} + \sqrt { - \tfrac{4ib}{{a\sqrt {\tfrac{A}{D} + \tfrac{B}{C} - \tfrac{2}{3a}} }} - \tfrac{B}{C} - \tfrac{A}{D} - \tfrac{4}{3a}} } \right)t}} , \\ \end{aligned}$$
(41)

where \(c_{i} ,i = 1,2, \ldots\) are constants of integrations and

$$A = \sqrt[3]{2}\left( {12ac + 1} \right),$$
$$B = \sqrt[3]{{ - 108ab^{2} - 72ac + \sqrt {\left( { - 108ab^{2} - 72ac + 2} \right)^{2} - 4\left( {12ac + 1} \right)^{3} + 2} }},$$
$$C = 3\sqrt[3]{2}a,$$
$$D = 3aB,$$

with \(a = {{\tau^{2} } \mathord{\left/ {\vphantom {{\tau^{2} } 4}} \right. \kern-0pt} 4}\), \(b = \omega \sin \lambda\) and \(c = \omega_{0}^{2}\). Since \(\omega ,\tau < < 1\), then \(ab^{2} < < 1\), \(\tau^{2} \omega_{0}^{2} < < 1\) and the previous parameters may be simplified respectively to: \(A \approx \sqrt[3]{2}\), \(B \approx \sqrt[3]{\sqrt 2 }\), \(C \approx {{3\sqrt[3]{2}\tau^{2} } \mathord{\left/ {\vphantom {{3\sqrt[3]{2}\tau^{2} } 4}} \right. \kern-0pt} 4}\) and \(D \approx {{3\sqrt[3]{\sqrt 2 }\tau^{2} } \mathord{\left/ {\vphantom {{3\sqrt[3]{\sqrt 2 }\tau^{2} } 4}} \right. \kern-0pt} 4}\). It is easy to check that Eq. (41) may be approximated by \(Q(t) \approx e^{{{{1.65it} \mathord{\left/ {\vphantom {{1.65it} \tau }} \right. \kern-0pt} \tau }}} \sin ({{\sqrt 3 t} \mathord{\left/ {\vphantom {{\sqrt 3 t} {2\tau }}} \right. \kern-0pt} {2\tau }})\) and therefore the nonlocal precession frequency is \(\approx {{1.65} \mathord{\left/ {\vphantom {{1.65} \tau }} \right. \kern-0pt} \tau }\) and the corresponding period is \(T = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \omega }} \right. \kern-0pt} \omega } \approx 3.8\tau\) which is very small compared to the standard result. The following 2nd statement holds as well.

Statement 2

For the case of the Foucault pendulum, the nonlocal precession frequency is \(\approx {{1.65} \mathord{\left/ {\vphantom {{1.65} \tau }} \right. \kern-0pt} \tau }\) and the corresponding period is too small compared to the standard result which is about \(30\tfrac{1}{2}\) at \(\lambda = 52^{ \circ } {\text{N}}\) . Consequently the nonlocal effect is insignificant in the case of the simple pendulum dynamics.

We plot in Figs. 18 and 19 the 3D variations of \(Q(t) \approx e^{{{{1.65it} \mathord{\left/ {\vphantom {{1.65it} \tau }} \right. \kern-0pt} \tau }}} \sin ({{\sqrt 3 t} \mathord{\left/ {\vphantom {{\sqrt 3 t} {2\tau }}} \right. \kern-0pt} {2\tau }})\) for specific ranges of time and \(\tau\):

Fig. 18
figure 18

General real part of \(Q(t)\)

Fig. 19
figure 19

General imaginary part of \(Q(t)\)

Although the nonlocal-in-time effect is extremely tiny at large time, there is an indication that disordered dynamics occurred at small range of time.

2.3

We discuss finally the problem of a mass \(m\) moving in a rotating tube assumed to be in the \((x,y)\) plane. The angular vector velocity is assumed to be \(\vec{\varOmega } = \varOmega \hat{z}\) and the nonlocal equation of motion is obtained from Eq. (7) and takes the form (El-Nabulsi 2015; Pardy 2011):

$$\frac{1}{4}\tau^{2} \vec{r}^{(4)} + \ddot{\vec{r}} = \vec{\varOmega }\left( {\vec{r} \times \vec{\varOmega }} \right),$$
(42)

The force is perpendicular to the angular velocity \(\varOmega\) and to the radius vector \(\vec{r}\). It is straightforward to check that Eq. (42) is reduced to:

$$\frac{1}{4}\tau^{2} r^{(4)} + \ddot{r} - \varOmega^{2} r = 0,$$
(43)

For simplicity we set \(\varOmega = 1\). Assuming initial conditions \(r(0) = 1\) and \(\dot{r}(0) = \ddot{r}(0) = \dddot r(0) = 0\), the solution of Eq. (43) is given by:

$$r(t) \approx 0.5\left( {e^{t} + e^{ - t} } \right) + 0.002\cos \left( {20.025t} \right),$$
(44)

for \(\tau = 10^{ - 1} \,{\text{s}}\) and by:

$$r(t) \approx 0.5\left( {e^{t} + e^{ - t} } \right) + 0.00002\cos \left( {200.002t} \right) + 1.38 \times 10^{ - 23} \sin \left( {200.002t} \right).$$
(45)

The standard solution is in fact given by \(r(t) \approx 0.5(e^{t} + e^{ - t} )\) and its variations with respect to Eqs. (44) and (45) are plotted in Figs. 20 and 21 for small and large ranges of time.

Fig. 20
figure 20

Variations of \(r(t)\) for \(\tau = 0\)(yellow curve), \(\tau = 0.1\,{\text{s}}\)(red curve) and \(\tau = 0.1\,{\text{s}}\)(blue curve) (small range of time)

Fig. 21
figure 21

Variations of \(r(t)\) for \(\tau = 0\)(yellow curve), \(\tau = 0.1\,{\text{s}}\)(red curve) and \(\tau = 0.1\,{\text{s}}\)(blue curve) (large range of time)

Perceptibly, the behavior of \(r(t)\) differs merely at small range of time. If the tune is joined with the North Pole of the Earth where \(\varOmega = 1/{\text{day}}\) and \(t = 1{\text{ day}}\), then Eq. (44) gives \(r \approx 1.54385\,{\text{m}}\), Eq. (45) gives \(r \approx 1.54308\,{\text{m}}\) whereas the standard result is \(r \approx 1.54308\,{\text{m}}\). It is easy to check that for \(\tau = 0.5\,{\text{s}}\) we find \(r \approx 1.40111\,{\text{m}}\). The following statement holds accordingly.

Statement 3

The nonlocality-in-time can be evaluated by the experiment with the rotating tube joined with the North Pole of the Earth. For \(\tau = 10^{ - 1} \,{\text{s}}\), the nonlocality effect represents about \(0.6\%\) whereas for \(\tau = 0.5\,{\text{s}}\), the nonlocality effect represents about \(9\%\).

3 Governing Nonlocal Equations of Fluid Dynamics Under the Influence of Earth Rotation

It is well-known that the dynamics of climate systems and ocean currents is considerably slow compared to the rotation speed of the Earth. This will put the Coriolis force as a most important physical ingredient in their dynamics. In this section, we shall investigate the dynamics of fluids in the Earth rotating reference system. We start by writing the Navier–Stokes equation in the Earth frame of reference which is rotating at a constant angular velocity \(\varOmega\) with respect to an inertial frame. From kinetic consideration and in the absence of jerks and hyperjerk, the acceleration in the absolute or inertial frame is connected to the acceleration in the rotating frame through the relation:

$$\ddot{\vec{r}}_{inertial} = \ddot{\vec{r}}_{rotational} + 2\vec{\varOmega } \times \dot{\vec{r}} + \vec{\varOmega } \times \vec{\varOmega } \times \vec{r},$$
(46)

and since in the inertial frame the Newton’s law \(\vec{F} = m\ddot{\vec{r}}_{inertial}\) holds, then in the rotating frame we can write:

$$\ddot{\vec{r}}_{rotational} = \frac{{\vec{F}}}{m} - 2\vec{\varOmega } \times \dot{\vec{r}} - \vec{\varOmega } \times \vec{\varOmega } \times \vec{r}.$$
(47)

For the case of ideal and incompressible fluid with pressure \(p\) and density \(\rho\) and in the absence of viscosity, we can replace the force by \(\vec{F} = - ({m \mathord{\left/ {\vphantom {m \rho }} \right. \kern-0pt} \rho })\nabla \vec{p} - mg\hat{z}\) and besides we can replace the total derivative by its Eulerian form \({d \mathord{\left/ {\vphantom {d {dt}}} \right. \kern-0pt} {dt}} = {\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-0pt} {\partial t}} + \vec{v} \cdot \nabla\) and therefore the Navier–Stokes equations in the rotating frame is:

$$\frac{{\partial \vec{v}}}{\partial t} + \vec{v} \cdot \nabla \vec{v} = - \frac{1}{\rho }\nabla \vec{p} - g\hat{z} - 2\vec{\varOmega } \times \vec{v} - \vec{\varOmega } \times \vec{\varOmega } \times \vec{r}.$$
(48)

However, in Suykens’s approach, the force in the inertial frame is obtained from Eq. (4) and takes the general form (Suykens 2009):

$$\vec{F} = m\ddot{\vec{r}}_{inertial} + m\sum\limits_{k = 1}^{n} {\frac{1}{4}\left( {1 - \left( { - 1} \right)^{k + 1} } \right)\left( {1 + \left( { - 1} \right)^{k} } \right)} \frac{{\tau^{k} }}{k!}\vec{r}_{inertial}^{(k + 2)} .$$
(49)

Therefore for \(n = 2\), we obtain:

$$\vec{F} = m\ddot{\vec{r}}_{inertial} + m\frac{{\tau^{2} }}{2}\vec{r}_{inertial}^{(4)} ,$$
(50)

and the resulting nonlocal-in-time Navier–Stokes equation in the rotating frame is:

$$\ddot{\vec{r}}_{rotational} + \frac{{\tau^{2} }}{2}\vec{r}_{rotational}^{(4)} = - \frac{{\nabla \vec{p}}}{\rho } - g\hat{z} - 2\vec{\varOmega } \times \dot{\vec{r}} - \vec{\varOmega } \times \vec{\varOmega } \times \vec{r} - \frac{{\tau^{2} }}{2}\left( {2\vec{\varOmega } \times \dddot{\vec{r}} + \vec{\varOmega } \times \vec{\varOmega } \times \ddot{\vec{r}}} \right).$$
(51)

Using \({d \mathord{\left/ {\vphantom {d {dt}}} \right. \kern-0pt} {dt}} = {\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-0pt} {\partial t}} + \vec{v} \cdot \nabla\), we can write Eq. (51) as:

$$\frac{{\partial \vec{v}}}{\partial t} + \vec{v} \cdot \nabla \vec{v} + \frac{{\tau^{2} }}{2}\left( {\frac{{\partial \vec{J}}}{\partial t} + \vec{v} \cdot \nabla \vec{J}} \right) = - \frac{{\nabla \vec{p}}}{\rho } - g\hat{z} - 2\vec{\varOmega } \times \vec{v} - \vec{\varOmega } \times \vec{\varOmega } \times \vec{r} - \frac{{\tau^{2} }}{2}\left( {2\vec{\varOmega } \times \vec{J} + \vec{\varOmega } \times \vec{\varOmega } \times \vec{a}} \right),$$
(52)

where \(\vec{J}\) is the jerk and \(\vec{a}\) is the acceleration.

Remark 3.1

The equations of hydrodynamics are merely an approximation of their kinetic analogues-which are more fundamental. Accordingly, addition of a new mathematical feature in the hydrodynamical equations is physically justified if it originates from a parallel development at the kinetic level. So one naturally asks me how Eq. (52) originates from the Boltzmann equation. In fact, according to the Boltzmann’s approach, if particles are subject to an external force \(\vec{F}\) not due to other particles, the probability density function \(f(\vec{x},\vec{p},t)\) satisfies in the presence of collisions the following partial differential equation:

$$\frac{\partial f}{\partial t} + \frac{{\vec{p}}}{m} \cdot \nabla_{{\vec{x}}} f + \vec{F} \cdot \nabla_{{\vec{p}}} f = \left( {\frac{\partial f}{\partial t}} \right)_{\text{collisions}} .$$

Here the function \(f(\vec{x},\vec{p},t):{\mathbb{R}}^{3} \times {\mathbb{R}}^{3} \times {\mathbb{R}}^{ + } \to {\mathbb{R}}^{ + }\)(\((t,\vec{x},\vec{p}) \in [0, + \infty [ \times {\mathbb{R}}^{3} \times {\mathbb{R}}^{3}\), \(\vec{x} \in {\mathbb{R}}^{3}\) and \(\vec{p} \in {\mathbb{R}}^{3}\) are position and momentum respectively (Harris 2004). This equation may be simply written as \(\hat{L}[f] = C[f]\) where \(C\) is the collision operator and \(\hat{L}\) is the Liouville operator defined by:

$$\hat{L} = \frac{\partial }{\partial t} + \frac{{\vec{p}}}{m} \cdot \nabla_{{\vec{x}}} + \vec{F} \cdot \nabla_{{\vec{p}}} .$$

In fact, this formulation is nothing a mathematical approximation and fails to accurately describe the collisional dynamics of particles the reason physicists use in general the BBGKY hierarchy of equations which involve all distribution functions concurrently. One may to implement nonlocality in Boltzmann theory is to replace the Liouville operator by El-Nabulsi (2018):

$${\hat{L}} = \frac{\partial }{\partial t} + \left( {\frac{{\vec{p}}}{m} + \frac{{\tau^{2} }}{2}\vec{J}} \right) \cdot \nabla_{{\vec{x}}} + \left( {\vec{F} + \frac{{\tau^{2} }}{2}m\vec{S}} \right) \cdot \nabla_{{\vec{p}}} ,$$
$$\equiv \hat{L} + \frac{{\tau^{2} }}{2}\left( {\vec{J} \cdot \nabla_{{\vec{x}}} + m\vec{S} \cdot \nabla_{{\vec{p}}} } \right),$$

which takes nonlocal effects. Here \(J \equiv x^{(3)}\) is again the jerk and \(S \equiv x^{(4)}\) is the snap. It was revealed in El-Nabulsi (2018) that this formalism may be used to rig the Boltzmann equation to ensure that it entails the desired form of the Navier–Stokes equation.

In the Cartesian frame \((x,y,z)\) we have \(\vec{\varOmega } = (\varOmega_{x} ,\varOmega_{y} ,\varOmega_{z} ) = (0,\varOmega \cos \phi ,\varOmega \sin \phi )\), \(\phi\) being the latitude and \(\varOmega\) the magnitude of the rotation vector. We can define a nonlocal effective gravity term by:

$$-\,g_{effective} \hat{\tilde{z}} = - \,g\hat{z} - \vec{\varOmega } \times \vec{\varOmega } \times \vec{r} - \frac{{\tau^{2} }}{2}\vec{\varOmega } \times \vec{\varOmega } \times \vec{a},$$
(53)

where \(-\, g\hat{z}\) is the true gravity and points away from the Earth’s center and the vector \(\hat{\tilde{z}}\) is the true local vertical (Huang 2011). It is notable that the nonlocal centrifugal force \(-\, m\vec{\varOmega } \times \vec{\varOmega } \times \vec{r} - m{{(\tau^{2} } \mathord{\left/ {\vphantom {{(\tau^{2} } 2}} \right. \kern-0pt} 2})\vec{\varOmega } \times \vec{\varOmega } \times \vec{a}\) can generally be ignored at the surface of the Earth. Using the effective gravity in Eq. (52), it is straightforward to verify that the components of the nonlocal-in-time Navier–Stokes equation are:

$$\begin{aligned} & \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} + \frac{{\tau^{2} }}{2}\left( {\frac{{\partial J_{u} }}{\partial t} + u\frac{{\partial J_{u} }}{\partial x} + v\frac{{\partial J_{u} }}{\partial y} + w\frac{{\partial J_{u} }}{\partial z}} \right) \\ & = - \frac{1}{\rho }\frac{\partial p}{\partial x} - 2w\varOmega \cos \phi + 2v\varOmega \sin \phi + \tau^{2} \left( { - J_{w} \varOmega \cos \phi + J_{v} \varOmega \sin \phi } \right), \\ \end{aligned}$$
(54)
$$\begin{aligned} & \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z} + \frac{{\tau^{2} }}{2}\left( {\frac{{\partial J_{v} }}{\partial t} + u\frac{{\partial J_{v} }}{\partial x} + v\frac{{\partial J_{v} }}{\partial y} + w\frac{{\partial J_{v} }}{\partial z}} \right) \\ & = - \frac{1}{\rho }\frac{\partial p}{\partial y} - 2u\varOmega \text{cos} \phi - \tau^{2} J_{u} \varOmega \sin \phi , \\ \end{aligned}$$
(55)
$$\begin{aligned} & \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z} + \frac{{\tau^{2} }}{2}\left( {\frac{{\partial J_{w} }}{\partial t} + u\frac{{\partial J_{w} }}{\partial x} + v\frac{{\partial J_{w} }}{\partial y} + w\frac{{\partial J_{w} }}{\partial z}} \right) \\ & = - \frac{1}{\rho }\frac{\partial p}{\partial z} - g + 2u\varOmega \cos \phi + \tau^{2} J_{u} \varOmega \cos \phi , \\ \end{aligned}$$
(56)

where \(\vec{v} = (u,v,w)\) and \(\vec{J} = (J_{u} ,J_{v} ,J_{w} )\). However, for large-scale atmospheric dynamics, \(\left| {w\varOmega \cos \phi } \right| < < \left| {v\varOmega \cos \phi } \right|\) and consequently \(\left| {J_{w} \varOmega \cos \phi } \right| < < \left| {J_{v} \varOmega \cos \phi } \right|\), i.e. the horizontal velocity is much greater than the vertical velocity and consequently the horizontal jerk is much greater than the vertical jerk. Besides, the horizontal pressure gradient force and the horizontal component of Coriolis force are roughly equal (in magnitude). Since we have neglected the centrifugal acceleration the equations of motion as well as the boundary conditions are independent of the exact position in the \((x,y)\) plane. Moreover, the vertical velocity is a constant since the mass conservation implies \({{\partial w} \mathord{\left/ {\vphantom {{\partial w} {\partial z}}} \right. \kern-0pt} {\partial z}} = 0\), i.e. since at \(z = 0\) we must have \(w = 0\), then \(w = 0\) ubiquitously (Ekman layer). Since hydrostatic balance holds to a high degree of accuracy in the vertical direction, we can simplify Eqs. (54)–(56) respectively to:

$$\frac{du}{dt} + \frac{{\tau^{2} }}{2}\frac{{dJ_{u} }}{dt} = - \frac{1}{\rho }\frac{\partial p}{\partial x} + \left( {v + \frac{{\tau^{2} }}{2}J_{v} } \right)\xi ,$$
(57)
$$\frac{dv}{dt} + \frac{{\tau^{2} }}{2}\frac{{dJ_{v} }}{dt} = - \frac{1}{\rho }\frac{\partial p}{\partial y} - \left( {u + \frac{{\tau^{2} }}{2}J_{u} } \right)\xi ,$$
(58)
$$0 = - \frac{1}{\rho }\frac{\partial p}{\partial z} - g,$$
(59)

where \(\xi = 2\varOmega \sin \phi\) is the Coriolis parameter. Neglecting the meridional excursion and the horizontal gradient pressure of the fluid, we can set \(\xi = \xi_{0}\)(constant) and therefore Eqs. (57) and (58) are simplified respectively to:

$$\frac{du}{dt} + \frac{{\tau^{2} }}{2}\frac{{dJ_{u} }}{dt} = \left( {v + \frac{{\tau^{2} }}{2}J_{v} } \right)\xi_{0} ,$$
(60)
$$\frac{dv}{dt} + \frac{{\tau^{2} }}{2}\frac{{dJ_{v} }}{dt} = - \left( {u + \frac{{\tau^{2} }}{2}J_{u} } \right)\xi_{0} ,$$
(61)

which are combined into:

$$\frac{{d^{2} W}}{{dt^{2} }} + \frac{{\tau^{2} }}{2}\frac{{d^{4} W}}{{dt^{2} }} + \xi_{0} \left( {W + \frac{{\tau^{2} }}{2}\frac{{d^{2} W}}{{dt^{2} }}} \right) = 0,$$
(62)

where \(W = (u,v)\). The solution of Eq. (62) is given by:

$$\begin{aligned} W(t) & = c_{5} e^{{ - \tfrac{t}{2}\sqrt { - \tfrac{{2(\xi_{0} \tau^{2} + 4)}}{{\tau^{2} }} - 2\sqrt {\tfrac{{(\xi_{0} \tau^{2} + 4)^{2} }}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} + c_{6} e^{{\tfrac{t}{2}\sqrt { - \tfrac{{2(\xi_{0} \tau^{2} + 4)}}{{\tau^{2} }} - 2\sqrt {\tfrac{{(\xi_{0} \tau^{2} + 4)^{2} }}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} \\ & \quad + c_{7} e^{{ - \tfrac{t}{2}\sqrt { - \tfrac{{2(\xi_{0} \tau^{2} + 4)}}{{\tau^{2} }} + 2\sqrt {\tfrac{{(\xi_{0} \tau^{2} + 4)^{2} }}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} + c_{8} e^{{\tfrac{t}{2}\sqrt { - \tfrac{{2(\xi_{0} \tau^{2} + 4)}}{{\tau^{2} }} + 2\sqrt {\tfrac{{(\xi_{0} \tau^{2} + 4)^{2} }}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} . \\ \end{aligned}$$
(63)

To illustrate numerically we set \(f_{0} = 1\) (for simplicity) and we plot in Figs. 22, 23, 24, 25 the variations of Eq. (63) for \(\tau = 0.1\) and for different initial conditions.

Fig. 22
figure 22

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (1,0,0,0)\)

Fig. 23
figure 23

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,1,0,0)\)

Fig. 24
figure 24

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,1,0)\)

Fig. 25
figure 25

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,0,1)\)

Figures 26, 27, 28, 29 illustrate the variations of Eq. (63) for \(\tau = 0.01\) and for different initial conditions:

Fig. 26
figure 26

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (1,0,0,0)\)

Fig. 27
figure 27

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,1,0,0)\)

Fig. 28
figure 28

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,1,0)\)

Fig. 29
figure 29

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,0,1)\)

Finally, Figs. 30, 31, 32, 33 illustrate the variations of Eq. (63) for \(\tau = 0.5\) and for different initial conditions.

Fig. 30
figure 30

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (1,0,0,0)\)

Fig. 31
figure 31

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,1,0,0)\)

Fig. 32
figure 32

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,1,0)\)

Fig. 33
figure 33

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,0,1)\)

And finally, Figs. 34, 35, 36, 37 illustrate the variations of Eq. (63) for \(\tau = 0.25\) and for different initial conditions.

Fig. 34
figure 34

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (1,0,0,0)\)

Fig. 35
figure 35

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,1,0,0)\)

Fig. 36
figure 36

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,1,0)\)

Fig. 37
figure 37

Variations of \(W(t)\) for \((W,\dot{W},\ddot{W},\dddot W)(0) = (0,0,0,1)\)

We observe that the nonlocal dynamics is extremely sensitive for initial conditions and for values of the nonlocal parameter \(\tau\). The phase-space trajectories show dissimilar beat frequencies of the fluid system. For special initial conditions, the motion is roughly periodic but exhibits to some extent a disordered behavior.

Remark 3.2

It should be noted that the observed sensitivity is higher for the second and third derivative of \(W(t)\) than for the other part \(W(t)\) and \(\dot{W}(t)\) due to nonlocality of the dynamics (see El-Nabulsi 2017).

Remark 3.3

In fact, since \(\xi_{0} \tau^{2} < < 1\), Eq. (62) may be simplified to:

$$\frac{{\tau^{2} }}{2}\frac{{d^{4} W}}{{dt^{2} }} + \frac{{d^{2} W}}{{dt^{2} }} + \xi_{0} W = 0,$$

and therefore Eq. (63) is simplified to:

$$W(t) = c_{9} e^{{ - \tfrac{t}{2}\sqrt { - \tfrac{8}{{\tau^{2} }} - 2\sqrt {\tfrac{16}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} + c_{10} e^{{\tfrac{t}{2}\sqrt { - \tfrac{8}{{\tau^{2} }} - 2\sqrt {\tfrac{16}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} + c_{11} e^{{ - \tfrac{t}{2}\sqrt { - \tfrac{8}{{\tau^{2} }} + 2\sqrt {\tfrac{16}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} + c_{12} e^{{\tfrac{t}{2}\sqrt { - \tfrac{8}{{\tau^{2} }} + 2\sqrt {\tfrac{16}{{\tau^{4} }} - \tfrac{{16\xi_{0} }}{{\tau^{2} }}} } }} .$$

One may check that the numerical solutions are roughly similar to those illustrated in Figs. (22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37).

One argues that the dynamics is affected by the presence of nonlocality-in-time and there is a transition from order to disorder. The following statement holds consequently.

Statement 4

Under the influence of Earth rotation and for large atmospheric dynamics, neglecting the meridional excursion and the horizontal gradient pressure of the fluid, the nonlocal-in-time NavierStokes equations in rotating frame results on a disordered dynamics which is extremely sensitive for initial conditions and for values of the nonlocal parameter \(\tau\).

4 Conclusions and Perspectives

In this work, we have studied the nonlocal-in-time dynamics in the rotating frame based on nonlocal-in-time kinetic energy approach recently introduced by Suykens. Our main aim was to perceive the impacts of nonlocality on large rotating dynamics like the Earth. We have observed that nonlocality-in-time has an effect on the dynamics of bodies in a rotating frame and besides it can be measured as proved in the free-falling problem mainly on large spatial and small ranges of time. For the case of the Foucault simple pendulum, the nonlocal effect is insignificant since its associated period is proportional to the nonlocal parameter \(\tau\). The motion of a massive body in a rotating tube showed in contrast that the nonlocal effect is significant. We have studied the dynamics of fluids in the Earth rotating reference system after constructing the nonlocal-in-time Navier–Stokes equations in the Earth frame of reference. For large atmospheric dynamics, it was observed that a transition from order to disorder occurs and that the dynamics is extremely sensitive for initial conditions and for the values of \(\tau\). The main conclusion of this paper is that nonlocality has significant effects on classical laws of fluid dynamics even at large spatial scale and which could be measured experimentally. It will be interesting to explore the impacts of nonlocality-in-time in strongly turbulent media like the Earth’s oceans and atmospheres among them being Rossby waves (Cushman-Roisin and Beckers 2011). Works in this direction are under construction.