# Equatorial magnetohydrodynamic shallow water waves in the solar tachocline

###### Abstract

The influence of toroidal magnetic field on the dynamics of shallow water waves in the solar tachocline is studied. Sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline causes significant reduction of surface gravity speed, which leads to the trapping of the waves near the equator and to the increase of Rossby wave period up to the time scale of solar cycles. Dispersion relations of all equatorial MHD shallow water waves are obtained in the upper tachocline conditions and solved analytically and numerically. It is found that the toroidal magnetic field splits equatorial Rossby and Rossby-gravity waves into fast and slow modes. For reasonable value of reduced gravity, global equatorial fast magneto-Rossby waves (with the spatial scale of equatorial extent) has the periodicity of 11 yrs matching the time scale of activity cycles. The solutions are confined around the equator between latitudes coinciding with sunspot activity belts. Equatorial slow magneto-Rossby waves have the periodicity of 90-100 yrs resembling observed long-term modulation of cycle strength i.e. Gleissberg cycle. Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves have the periodicity of 1-2 yrs and may correspond to observed annual and quasi-biennial oscillations. Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves have the period of hundreds of days and might be responsible for observed Rieger-type periodicity. Consequently, the equatorial MHD shallow water waves in the upper overshoot tachocline may capture all time scales of observed variations in solar activity, but detailed analytical and numerical studies are necessary to make firm conclusion towards the connection of the waves to solar dynamo.

^{†}

^{†}journal: ApJ

## 1 Introduction

Rossby waves govern large scale dynamics of Earth’s atmosphere and oceans (Rossby, 1939; Gill, 1982). The waves arise owing to the conservation of absolute vorticity and have been also considered to have important influence on dynamics of astrophysical discs (Lovelace and Hohlfeld, 1978; Lovelace et al., 1999; Umurhan, 2010), solar-like stars (Lanza et al., 2009; Bonomo and Lanza, 2012), neutron stars (Andersson, 1999; Lou, 2001; Heng and Spitkovsky, 2009), planetary (Petviashvili, 1980) and exo-planetary atmospheres (Heng and Workman, 2014). Rossby waves have been used to explain the observed mid-range (or Rieger-type) periodicity in solar activity, which is seen in many activity indices such as sunspot number, radio flux, flares and CMEs (Rieger et al., 1984; Carbonell & Ballester, 1990; Oliver et al., 1998). Recent direct observation of Rossby waves in coronal bright points (McIntosh et al., 2017) confirmed their important role in large scale dynamics of solar atmosphere and interior.

Coriolis parameter, which is actually the vorticity of rotating sphere, depends on latitude with minimum at the equator. Therefore, the equatorial region serves as a cavity for various shallow water waves (in analogy of quantum harmonic oscillator, where potential walls trap oscillations). These so called equatorially trapped or equatorial waves have been intensively studied by Matsuno (1966), Longuet-Higgins (1968) and more recently by Bouchut et al. (2005). Lou (2000) suggested that the equatorially trapped Rossby and inertia-gravity waves in the solar photosphere may explain Rieger type periodicity in solar activity. Consideration of Lou (2000) is based on hydrodynamic (HD) shallow water equations, while observed strong magnetic field on the Sun naturally supposes to consider magnetohydrodynamic (MHD) effects.

Gilman (2000) presented MHD shallow water equations for horizontal magnetic field, which have been used for various applications to the solar tachocline (Dikpati and Gilman, 2001; Dikpati et al., 2003, 2006, 2007; Zaqarashvili et al., 2010a, b, 2015). Schecter et al. (2001) studied various MHD shallow water waves in -plane approximation in the solar tachocline, but they did not consider equatorially trapped waves. Zeitlin (2013) studied shallow water MHD waves in - and -planes using quasi-geostrophic approximation and found that the magnetic field has a stabilising effect on the baroclinic instability. Dikpati and Gilman (2001) and Dikpati et al. (2003, 2006, 2007) performed detailed stability analysis of shallow water system under the actions of the horizontal magnetic field with various latitudinal profiles and the latitudinal differential rotation. It was found that the joint action of the differential rotation and the horizontal magnetic field generally leads to the instability of shallow water system. Nonlinear parametric interaction may also cause the mutual energy transformation between various waves in MHD shallow water approximation (Klimachkov and Petrosyan, 2017).

Despite of significant work based on MHD shallow water approximation, the linear spectrum of equatorial waves was not studied in details in the tachocline with horizontal magnetic field. This consideration might have remarkable importance for two main reasons. First, the magnetic field may significantly influence the spectrum of HD shallow water waves leading to e.g. their splitting (Zaqarashvili et al., 2007, 2009; Heifetz et al., 2015). Second, sub-adiabatic temperature gradient in the upper overshoot part of the tachocline creates negative buoyancy force leading to the reduced gravity (Gilman, 2000; Dikpati and Gilman, 2001), which may cause significant increase of Rossby wave time scales up to the periodicity of solar cycles.

Here we use the MHD shallow water equations and study the spectrum of equatorial waves in the solar tachocline.

## 2 Governing equations

We consider the solar tachocline as a shallow layer with thickness ( cm) located at the distance ( cm) from the solar center (Spiegel and Zahn, 1992). We consider a local Cartesian frame on rotating Sun, where is directed towards west (i.e. in the direction of solar rotation), is directed towards north, and is directed vertically outwards. The layer is perceived with unperturbed toroidal magnetic field which is generally latitude-dependent - .We adopt solid body rotation with the angular velocity - =2.610 s. Differential rotation is neglected at this stage for two reasons. First, the value of differential rotation is small near the equator. Second, the differential rotation is of importance for instabilities and it will not significantly affect wave dispersion relations.

We start with linear shallow water MHD equations in rotating Cartesian frame (Gilman, 2000)

(1) |

(2) |

(3) |

(4) |

(5) |

where and are the velocity perturbations, and are the magnetic field perturbations, is the perturbation of layer thickness, is the reduced gravity, is the fluid density and is the Coriolis parameter with being a latitude. Here we use Cartesian coordinates, which capture essential dynamics of large-scale equatorially trapped waves (Matsuno, 1966; Lou, 2000). The Cartesian coordinates allow us more easier treatment of wave dispersion relations which are similar to those obtained under spherical geometry (Matsuno, 1966; Longuet-Higgins, 1968).

Reduced gravity, which is an essential part of shallow water system in the tachocline, is owing to the sub-adiabatic temperature gradient providing a negative buoyancy force to the deformed upper surface (Gilman, 2000). Therefore, the surface feels less gravitational field compared to the real gravity. Negative buoyancy force is proportional to the fractional difference between actual and adiabatic temperature gradients , which is in the range of in the upper overshoot part of the tachocline and may reach up to in the lower radiative part of the tachocline (Dikpati and Gilman, 2001). Dikpati and Gilman (2001) showed that the dimensionless value of reduced gravity is proportional to , therefore it is in the range of in the radiative part of the tachocline and in the range of in the upper overshoot part.

Differentiating Eqs. (1) and (2) by time and inserting time derivatives of , and from Eqs. (3)-(5) we arrive to the equations

(6) |

(7) |

where is the Alfvén speed and is the surface gravity speed. Fourier expansion of Eqs. (6)-(7) with , leads to the equation

(8) |

where sign shows differentiation by .

In the local frame at the latitude , Coriolis parameter can be expanded as . Retaining only the first order term in the expansion (so called -plane approximation) we get the Coriolis parameter near the equator as

(9) |

where (not to confuse with plasma )

(10) |

The structure and consequently solutions of Eq. (8) depend on the latitudinal structure of magnetic field.

## 3 Equatorial shallow water MHD waves in the solar tachocline with uniform toroidal magnetic field

Let us suppose that the magnetic field is uniform, i.e. . Then Eq. (8) is simplified to the equation

(11) |

where

(12) |

This is the equation of parabolic cylinder (also known as the equation of quantum harmonic oscillator) and when

(13) |

then it has bounded solutions (Abramowitz & Stegun, 1964)

(14) |

where is the Hermite polynomial of order and is a constant. The solutions are oscillatory inside the interval

(15) |

and exponentially tend to zero outside.

Eq. (15) shows that the waves are trapped near the equator for small and . Therefore, in order to have equatorially trapped shallow water waves one needs very small surface gravity speed i.e. very small reduced gravity, . This means that the waves can be trapped near the equator only in the upper overshoot part of the tachocline, where the reduced gravity is very small. In the lower radiative part of the tachocline the reduced gravity is still large, therefore the shallow water waves can not be trapped near the equator, but rather penetrate to higher latitudes (for the same reason the HD shallow water waves considered by Lou (2000) in the solar photosphere were not really concentrated near the equator but were extended up to latitudes). Therefore, in the following part of the paper, we will consider only the upper overshoot layer of the tachocline, where the reduced gravity is very small and hence it creates excellent conditions for the trapping of shallow water waves near the equator.

Eq. (13) defines the dispersion relation for equatorial MHD shallow water waves as

(16) |

When the magnetic field is zero, then the dispersion relation transforms into the dispersion relation of equatorial HD waves (Matsuno, 1966). This is the fourth order algebraic equation and its analytical solution is further complicated owing to the square root on the right hand-side of the equation. The numerical solution also needs certain caution as it may lead to spurious zeroes. The accepted strategy is that one should remove the square root by taking square of both side of the equation and find the solutions of resulted polynomial. This procedure obviously adds four additional solutions, which are not real. Therefore, obtained solutions should be carefully checked and verified if they satisfy the initial equation. After the careful check we obtained the numerical solutions of the dispersion relation in terms of different wave modes which are plotted on Figure 1 for positive frequency. This figure also shows the dispersion curves of HD shallow water waves denoted by dashed lines.

### 3.1 Magneto-inertia-gravity and magneto-Rossby waves for

For , the solutions include magneto-inertia-gravity and magneto-Rossby waves. The remarkable difference with regards to HD case is that the uniform toroidal magnetic field creates low frequency cut-off areas in the dispersion relation. The cut-off area is defined by lines (black dashed lines on Figure 1), which is also clearly seen in Eq. (16) as the right-hand side term becomes imaginary. This means that the low-frequency Rossby wave solution, which is plotted by red dashed line on Figure 1 in HD case, is not solution of dispersion relation anymore as it is prohibited by horizontal magnetic field. This can be understood physically as an action of Lorentz force on the vorticity of Rossby waves. Strong magnetic field opposes the vortex motion and prohibits the Rossby waves. As the magnetic field strength weakens then the low frequency Rossby waves may still arise. But the magnetic field with strength of 10 kG blocks the appearance of low frequency Rossby waves. The solution of magneto-Rossby waves is just above Alfvén wave solution . The phase and group speeds of the magneto-Rossby waves are directed eastward and hence the waves propagate opposite to the solar rotation. Magneto-inertia-gravity waves are both westward and eastward propagating (both, phase and group speeds are in the same direction) and they have the same dynamics as in HD case affected by the magnetic field mostly for higher wave numbers. For , which actually means pure poleward (or equator-ward) propagation, only magneto-inertia-gravity waves remain and their dispersion relation is .

#### 3.1.1 Magneto-inertia-gravity and magneto-Rossby-gravity waves for n=0

Eq. (16) needs special treatment for case and can be factorised for as

(17) |

The first expression of Eq. (17) has a zero when for positive . However, this solution is spurious as it leads to zero coefficients in front of velocity perturbations on the left hand side of Eq. (6)-(7). The second expression of Eq. (17) leads to the real solution for positive and negative (blue solid line on Figure 1). The solution for negative approaches the magneto-Rossby wave curve in high wavenumber limit. The phase and group speeds are directed eastward like magneto-Rossby waves, therefore this is a mixed magneto-Rossby-gravity mode. On the other hand, for the positive the solution approaches the magneto-inertia-gravity wave curve in high wavenumber limit with the westward directed phase and group speeds. However, there are two remarkable differences between the waves in MHD (blue solid line) and HD (blue dashed line) cases. First, the magnetic field creates a lower-frequency cut-off area also for mixed Rossby-gravity waves with negative wavenumber. Second, the group velocity of HD mixed Rossby-gravity wave is always westward, while the velocity becomes eastward for mixed magneto-Rossby-gravity waves.

### 3.2 Magneto-Kelvin waves

When poleward components of velocity () and magnetic field () are zero, then the particular class of solutions magneto-Kelvin waves arise similar to HD Kelvin waves. Eqs. (1)-(5) are now rewritten as

(18) |

(19) |

(20) |

(21) |

The Fourier expansion of Eqs. (18)-(20) with leads to the dispersion relation

(22) |

which yields two different modes. The solutions are

(23) |

for mode and

(24) |

for mode. It is seen that the first solution does not satisfy boundary condition, therefore it is ruled out from the consideration. Hence, only the mode with the dispersion relation

(25) |

remains as the solution for magneto-Kelvin waves (magenta line on Figure 1).

This solution also arises from general dispersion relation Eq. (16) if one substitutes . Therefore, this mode was called as mode by Matsuno (1966).

Figure 2 displays the dependence of shallow water wave periods on magnetic field strength. It is seen that magneto-Rossby and magneto-Kelvin waves yield the periods close to the Rieger periodicity for the field strength of 50 kG. The low-frequency cut-off is also seen (black line) for strong magnetic field, which is consequently removed for weaker field.

## 4 Equatorial shallow water MHD waves in the solar tachocline with nonuniform toroidal magnetic field

Now we consider latitudinally nonuniform toroidal magnetic field. Differential rotation inevitably leads to the nonuniform toroidal field in the case of any poloidal weak magnetic field.

Solar observed latitudinal differential rotation can be expressed as

(26) |

where and are parameters determined by observations whose values at the solar surface (and in the tachocline) are about 0.13-0.14.

The toroidal component of induction equation

(27) |

allows to derive the latitudinal profile of generated toroidal magnetic field as

(28) |

where is the time scale of the process. Therefore, the most obvious latitudinal profile of toroidal field is , which has been frequently used for the tachocline started by Gilman and Fox (1997).

This profile can be approximated near the equator as

(29) |

Inserting this expression into Eq. (8) and keeping only the terms with we get the equation

(30) |

By substitution of the expression

(31) |

this equation transforms into the standard form

(32) |

where

(33) |

When

(34) |

then Eq. (32) has the bounded solution

(35) |

where is the Hermite polynomial of order and is a constant. The solutions are oscillatory inside the interval

(36) |

and exponentially tend to zero outside. Eq. (34) defines the dispersion relation

(37) |

This is the six order polynomial equation with , therefore its exact analytical solution is very complicated also owing to the square root in right-hand side. Numerical solution also should be performed with caution (see the previous section). However, before starting the numerical solution, it is possible to find analytical solutions in some approximation. One can observe that is the singular point. It is seen from Eqs. (6)-(7) that this solution at the equator yields zero coefficients in front of velocity perturbations on the left-hand side (actually, the singular point corresponds to magneto-Kelvin waves as we will see later). Therefore, we can look to the approximate solutions for and .

### 4.1 Magneto-inertia-gravity waves ()

In this approximation, one can get the dispersion relation

(38) |

which is similar to the dispersion relation of HD shallow water waves and for week magnetic field approximation it completely transforms into HD inertia-gravity wave solution.

### 4.2 Magneto-Rossby waves ()

The other limit of smaller frequency waves () leads to the dispersion relation

(39) |

This is the second order polynomial with and it corresponds to magneto-Rossby waves. The magnetic field splits the equatorial HD Rossby waves into fast and slow magneto-Rossby waves (see also Zaqarashvili et al. (2007, 2009)). The dispersion relation for the high frequency fast modes can be further simplified to

(40) |

On the other hand, the dispersion relation for the low frequency slow modes can be simplified to

(41) |

The fast magneto-Rossby waves are similar to HD Rossby waves, while the slow magneto-Rossby wave is new type of waves which arise owing to both Coriolis and Lorentz forces.

### 4.3 Magneto-Kelvin waves

In the case of magnetic field profile (Eq. 29), Eqs. (18)-(21) are rewritten after the Fourier transformation as

(42) |

(43) |

(44) |

(45) |

Eqs. (42)-(44) lead to the dispersion relation

(46) |

which yields two different modes. The solutions for each mode can be easily found from Eqs. (44)-(45) for

(47) |

for mode and

(48) |

for mode. When , then the first solution satisfies the boundary conditions, hence it is the solution in weak magnetic field limit. However, for very strong magnetic field with the second solution satisfies the boundary conditions. For the magnetic field strength of 100 kG the mode with the dispersion relation

(49) |

remains as the solution for magneto-Kelvin waves. It is seen that the mode has different frequency at different layer. At the equator, its dispersion relation is

(50) |

similar to the HD case. It was expected since the magnetic field vanishes at .

Numerical solution of dispersion relation, Eq. (37), should be again performed with caution. We make square of both sides of the equation and solve resulted 12th order polynomial numerically. The spurious solutions are carefully determined and removed from the consideration. Resulted real solutions are plotted on Figure 3, which show the very different time scales of magneto-inertia-gravity and magneto-Rossby waves (note that corresponding HD wave modes are plotted by dashed lines). Mixed magneto-Rossby-gravity (=0) and magneto-Kelvin (=-1) waves have intermediate time scales. Magnetic field has almost no influence on the frequency of inertia-gravity and Kelvin waves, but it leads to the splitting of HD Rossby waves (red dashed line) into fast and slow magneto-Rossby waves (red solid lines). Westward propagating magneto-Rossby-gravity waves are not affected by the magnetic field. On the other hand, eastward propagating magneto-Rossby-gravity waves (with negative wavenumber) are split into fast and slow modes (blue solid lines) by the magnetic field for small and large wave lengths. The magnetic field also leads to the appearance of several cut-off areas. The first cut-off area appears for eastward magneto-Rossby-gravity waves in the wavelength interval of for the fixed parameters of magnetic field and reduced gravity. The second cut-off area arises for the fast magneto-Rossby-gravity waves in the wavelength interval of , where the mode crosses the dispersion curve (inclined black dashed line), which is spurious solution of the system. And the third, probably most important, cut-off area appears for fast magneto-Rossby and slow magneto-Rossby-gravity waves for large wavenumbers (vertical black dashed line). The phase speed of fast and slow magneto-Rossby waves is directed to the east i.e. the waves propagate in opposite direction of rotation. Their group speeds are also directed eastward. This is in contrast to HD Rossby waves, which have westward directed group speed for large wavelengths (Gill, 1982). Here the magnetic field blocks the westward group speed of Rossby waves. Fast and slow magneto-Rossby-gravity waves () propagate towards east, while magneto-inertia-gravity waves propagate towards west. For small wavelengths, the group speeds of fast and slow magneto-Rossby-gravity waves are directed westward and eastward, respectively. For large wavelengths, however, the group speed of fast magneto-Rossby-gravity waves is directed eastward and the group speed of slow magneto-Rossby-gravity waves is directed to the west. Magneto-inertia-gravity waves propagate towards east and west directions, while magneto-Kelvin waves propagate only to the west. The group speeds of both type of waves have the direction of their phase speeds.

The solutions of all shallow water waves are confined near the equator (for detailed spatial structures of HD equatorial waves see e.g. Bouchut et al. (2005)). Figure 4 shows the contour plots of harmonics of fast magneto-Rossby waves when unperturbed toroidal magnetic field () is 10 kG, normalised reduced gravity () is 0.001 and the normalised toroidal wave number () is 1. The harmonic is sandwiched between latitudes, has one zero (at the equator) as it is expected and consequently the opposite signs in the northern and southern hemispheres. The harmonic is sandwiched between latitudes and has two zeroes near latitudes.

Figure 5 shows the contour plots of all other variables in fast magneto-Rossby waves (with ) in the same parameters as in Figure 4. The perturbation of poleward magnetic field has the same spatial structure (lower right panel in Figure 5) as the poleward velocity (upper left panel in Figure 5), but with 180 degree shift in toroidal direction. This is clearly expected from Eq. (4). The solutions for the perturbations of layer thickness and toroidal velocity have the same sign in the two hemispheres. But they obviously change signs in the toroidal direction and have 180 degree phase shift.

The spitting of HD Rossby waves into fast and slow magneto-Rossby waves is displayed on Figure 6. When the wavelength reaches the toroidal extent of the equator (i.e. ) then the period of fast magneto-Rossby waves with is about 7 years for the field strength of 10 kG and the normalised reduced gravity of 0.001. For the wavelength the period is about 11 years exactly matching the solar cycle length. At the same time, the period of slow magnetic Rossby waves for reaches the value of 95 years, which is in the range of Gleissberg cycle (Gleissberg, 1939). The period of magneto-Kelvin waves is about 2 years, hence it is in the range of annual oscillations. Therefore, equatorial MHD shallow water waves cover almost all observed time scales of solar long-term activity variations in expected conditions of the overshoot layer.

## 5 Discussion

Solar activity undergoes variations over different time scales from tens/hundred days to tens/hundred years. Most pronounced variation occurs with a period of 11 yrs, which is usually explained by dynamo theory concerning differential rotation and convection. Besides the 11 yr cycles, several other periodicities are seen in different activity indices. Long-term modulation of cycle amplitude with the periodicity of 100 yrs (and more) is seen in sunspot numbers and cosmogenic radionuclides. On the other hand, shorter period variations are detected as annual (with periods of 1-2 yrs) and mid-range ( 200 days) oscillations. The physical mechanism(s) for long and short-term variations is not generally determined. On the other hand, Rossby wave scenario in the solar interior may capture essential physics of the variation.

Rossby (or planetary) waves owe their excitation to the conservation of absolute vorticity in rotating fluid. They lead to the formation of cyclones/anticyclones in higher latitudes of the Earth, which actually govern weather over Europe and USA. The waves are also trapped near the equator owing to the minimum of Coriolis parameter (so called equatorially trapped or equatorial waves), which might lead to the observed long period oscillations in oceans. Therefore, the Rossby waves generally determine the large-scale dynamics of the Earth’s atmosphere and oceans.

On the other hand, Rossby waves might be important for the large-scale dynamics of solar/stellar interior, where their properties are modified by large-scale magnetic field. The Rossby waves have been studied in the presence of the toroidal magnetic field and the differential rotation in the solar tachocline. Lou (2000) studied the dispersion relations for various equatorially trapped HD shallow water waves in the solar photosphere, but the dynamics of equatorial magneto-Rossby waves remained unexplored till now. In this paper, we studied the dynamics of equatorial shallow water waves in the solar tachocline for different latitudinal profiles of toroidal magnetic field. Initial MHD shallow water equations lead to the equation of parabolic cylinder near the equator, which has the solution in terms of Hermite polynomials satisfying the boundary conditions of equatorial confinement. The consideration allowed us to obtain the dispersion relations of various MHD shallow water waves for different magnetic field profiles.

First, equatorial HD shallow water waves previously described by Lou (2000) in the photosphere lead to essentially different periodicities in the upper overshoot tachocline (see dashed lines on Figures 1 and 3). The difference is related to the reduced gravity, which comes from the sub-adiabatic temperature gradient in the tachocline (Gilman, 2000; Dikpati and Gilman, 2001). The typical reduced gravity in the overshoot layer tends to the confinement of shallow water waves near the equator (the waves are extended up to 60 degree in the case of normal gravity) and leads to the significant increase of Rossby and Rossby-gravity wave period. The normal gravity considered by Lou (2000) in the photosphere yields the period of equatorial Rossby waves in the range of hundred days, while the typical reduced gravity ( 0.001) in the overshoot region yields the period of 7 yrs for the Rossby waves, hence approaching to the time scale of solar cycles in the case of 0.0004. On the other hand, the inertia-gravity waves show the Rieger-type periodicity of hundred days in the case of the reduced gravity.

Second (Section 3), the latitudinally uniform toroidal magnetic field has significant influence on the dispersion relations of equatorial shallow water waves. The main result of the influence is that the magnetic field creates low-frequency cut-off region forbidding the appearance of long-period Rossby and Rossby-gravity waves. The magnetic field with the strength of 10 kG blocks the appearance of low frequency Rossby waves owing to the action of the Lorentz force on the vorticity of Rossby waves. The magnetic field may also stabilise the baroclinic instability as recently studied by Zeitlin (2013). This point needs future detailed investigation also in the solar context. However, latitudinally uniform toroidal magnetic field probably is not a good approximation for the solar dynamo layer as the observed latitudinal differential rotation will inevitably lead to the nonuniform toroidal component.

Third (Section 4), the non-uniform toroidal magnetic field, which resembles solar magnetic field, affect the dispersion relations of shallow water waves in the tachocline conditions. We chose the latitudinal profile of for the toroidal magnetic field. This profile is actually obtained by action of observed differential rotation on a poloidal field component. It can be approximated as near the equator in Cartesian coordinate system. Sixth order polynomial dispersion equation (Eq. 37) is obtained, which describes all equatorial shallow water waves such as magneto-Rossby, magneto-Rossby-gravity, magneto-inertia-gravity and magneto-Kelvin waves. Simple analytical expressions and numerical solutions of general dispersion equation are derived, which show very interesting time-scales for all waves. The magnetic field splits the ordinary HD Rossby and Rossby-gravity waves into fast and slow modes, while the magneto-inertia-gravity waves are almost identical to their HD counterparts (see Figure 3). Magnetic field creates several cut-off areas for magneto-Rossby and magneto-Rossby-gravity waves. Most important cut-off area appears for large wavenumbers of fast magneto-Rossby and slow magneto-Rossby-gravity waves. The cut-off wavelength is 190 Mm for the magnetic field strength of 10 kG and reduced gravity of 0.001 (see Figure 3). It would be also interesting to study how the non-uniform magnetic field will influence on the baroclinic instability. We will briefly discuss each mode and corresponding observed time scale of solar activity variations.

### 5.1 Equatorial fast magneto-Rossby waves and Schwabe cycle

Solar activity undergoes 11-year oscillations known as Schwabe cycles (Schwabe, 1844). The Schwabe cycles are generally interpreted in terms of dynamo models, but the interpretation still has important problems (Charbonneau, 2005). Equatorial fast magneto-Rossby wave has similar time scale in certain conditions of overshoot layer. Figure 7 shows the dependence of magneto-Rossby, magneto-Rossby-gravity and magneto-Kelvin waves on the normalised value of reduced gravity, , for the field strength of 10 kG. We can see that the value of yields the period of fast magneto-Rossby waves with similar to the Schwabe cycles i.e. 11 years. The period of fast magneto-Rossby waves does not significantly depend on the strength of toroidal magnetic field (see Figure 8), therefore it is generally determined by the value of reduced gravity, which in its hand is owing to the sub-adiabatic temperature gradient.

The latitudinal extant of solutions depends on the normalised reduced gravity (lower latitudes with smaller ), therefore observed equator-ward drift of sunspots can be obtained if either temperature gradient or tachocline thickness is changing through the cycle. This can not be obtained by simple linear analysis considered in this paper but future nonlinear consideration my reveal the observed dynamics of sunspots.

### 5.2 Equatorial slow magneto-Rossby mode and Gleissberg cycle

Long-term records of sunspot number revealed the long-period modulation of Schwabe cycles with 80-100 yrs (Gleissberg, 1939; Hathaway, 2010). This modulation was explained in terms of slow magneto-Rossby waves excited in the lower part of the tachocline (Zaqarashvili et al., 2015). Figures 7-8 show that equatorial slow magneto-Rossby waves indeed lead to the observed modulation time scale. In contrast to the fast magneto-Rossby waves, the slow magneto-Rossby waves significantly depend on toroidal magnetic field strength. For the field strength of 10 kG, the period of slow magneto-Rossby waves tend to 90-100 yrs, which is in excellent coincidence with the Gleissberg cycle. The superposition of fast and slow magneto-Rossby modes may lead to the observed long-term modulation of activity cycles.

### 5.3 Equatorial magneto-Kelvin and slow magneto-Rossby-gravity waves vs annual/quasi-biennial oscillations

The oscillations with period 2 years (quasi-biennial oscillations) are found in almost all indices of solar activity (Sakurai, 1981; Vecchio et al., 2010). The oscillations were suggested to be connected with magneto-Rossby wave instability in the solar tachocline (Zaqarashvili et al., 2010b). Helioseismology also revealed the oscillations of solar tachocline velocity with the period of 1.3 years (Howe et al., 2000). Recent observations of coronal bright points based on STEREO and SDO data also showed annual oscillations (McIntosh et al., 2015), which were interpreted by Rossby waves in the solar interior (McIntosh et al., 2017). However, our results (see Figure 6-7) show that equatorial magneto-Kelvin and slow magneto-Rossby-gravity (=0) waves have time scale of 1-2 yrs, which coincide with observed annual oscillations. The magneto-Kelvin and magneto-Rossby-gravity waves do not significantly depend on the unperturbed magnetic field strength. Therefore, the value of normalised reduced gravity yields the period of magneto-Kelvin and slow magneto-Poincare-Rossby waves as 2 yrs for the wavenumber of (or ) and the period as 1 yr for the wavenumber of (or ). Again, there is a very nice coincidence with observed periodicity.

### 5.4 Equatorial fast magneto-Rossby-gravity and magneto-inertia-gravity waves vs Rieger-type periodicity

A short periodicity between 152–158 days was discovered in and X ray flares during solar cycle 21 by Rieger et al. (1984). Since the discovery, numerous papers observed the periodicity in different indices of solar activity (Carbonell & Ballester, 1990; Oliver et al., 1998; Gurgenashvili et al., 2016, 2017). The periodicity was explained by magneto-Rossby wave instability in the solar tachocline (Zaqarashvili et al., 2010a; Dikpati et al., 2017). However, our results indicate that in the framework of equatorial waves the periodicity is rather connected with magneto-inertia-gravity and/or fast magneto-Rossby-gravity waves than with magneto-Rossby waves. Figure 9 shows that eastward propagating fast magneto-Rossby-gravity wave and westward propagating magneto-inertia-gravity waves have the Rieger-type periodicity for the reduced gravity of .

## 6 Conclusion

We studied the linear dynamics of shallow water waves in the solar tachocline in the presence of toroidal unperturbed magnetic field. It is shown that the reduced gravity owing to the sub-adiabatic temperature gradient in the upper overshoot layer of the tachocline leads to the confinement of the waves near the equator. The dispersion relations of the equatorial waves are obtained for the latitudinally uniform and non-uniform toroidal magnetic field profiles. The dispersion relations are solved analytically and numerically describing the dynamics of various shallow water waves. Reasonable value of the temperature gradient in the overshoot region leads to the increase of the period of equatorial Rossby waves up to the time scale of solar cycles. Latitudinally uniform magnetic field creates the low-frequency cut-off region forbidding the occurrence of low-frequency Rossby and Rossby-gravity waves. Latitudinally non-uniform toroidal magnetic field (generated by observed latitudinal differential rotation from a weak poloidal component) leads to the both shorter and longer time scales of shallow water waves. The magnetic field splits ordinary HD Rossby waves into fast and slow magneto-Rossby modes and eastward propagating magneto-Rossby-gravity waves into fast and slow modes. It also blocks the appearance of fast magneto-Rossby and slow magneto-Rossby-gravity waves for large wavenumbers. On the other hand, the inertia-gravity and Kelvin waves are not significantly affected by the magnetic field. With reasonable value of reduced gravity in the overshoot region, the harmonics with of fast magneto-Rossby waves has the period of Schwabe cycles i.e. 11 years, which suggests a possible role of equatorial Rossby waves in generation of solar cycles. The solutions are concentrated between latitudes which coincide to the latitudes of sunspot appearance. In the same case, the period of equatorial slow magneto-Rossby waves is similar to the Gleissberg cycle i.e, 100 yrs. On the other hand, the periods of magneto-Kelvin (and slow magneto-Rossby-gravity) and magneto-inertia-gravity (and fast magneto-Rossby-gravity) waves correspond to observed annual/quasi-biennial and Rieger-type oscillations, respectively. Therefore, all modes of equatorial magneto shallow water waves reflect almost all observed periodicities in solar activity. Future analytical/numerical/observational study is surely required to settle the long-standing solar activity problem.

## References

- Abramowitz & Stegun (1964) Abramowitz, M., & Stegun, I.A. 1964, Handbook of Mathematical Functions (Washington, D.C.: National Bureau of Standards)
- Andersson (1999) Andersson, N., Kokkotas, K., and Schutz, B. F., 1999, ApJ, 510, 846
- Bonomo and Lanza (2012) Bonomo, A.S. and Lanza, A.F., 2012, å, 547, A37
- Bouchut et al. (2005) Bouchut, F., Le Sommer, J. and Zeitlin, V., 2005, Chaos, 15, 013503
- Carbonell & Ballester (1990) Carbonell, M., & Ballester, J.L. 1990, A&A, 238, 377
- Charbonneau (2005) Charbonneau, P. 2005, LRSP, 2, 2
- Dikpati and Gilman (1999) Dikpati, M., & Gilman, P. A. 1999, ApJ, 512, 417
- Dikpati and Gilman (2001) Dikpati, M., & Gilman, P. A. 1999, ApJ, 551, 536
- Dikpati et al. (2003) Dikpati, M., Gilman, P. A., & Rempel, M. 2003, ApJ, 596, 680
- Dikpati et al. (2006) Dikpati, M., Gilman, P. A. and MacGregor, K.B. 2006, ApJ, 638, 564
- Dikpati et al. (2007) Dikpati, M., Gilman, P. A., de Toma, G. and Ghosh, S. S., 2007, Solar Phys., 245, 1
- Dikpati et al. (2017) Dikpati, M., Cally, P. S., McIntosh, S. W., and Heifetz, E., 2017, Nature Rep., 7, 14750
- Gill (1982) Gill, A. E. 1982, Atmosphere-Ocean Dynamics, San Diego: Academic Press
- Gilman and Fox (1997) Gilman, P. A. and Fox, O. A., 1997, ApJ, 484, 439
- Gilman (2000) Gilman, P. A. 2000, ApJ, 544, L79
- Gleissberg (1939) Gleissberg, M. N., 1939, Observatory, 62, 158
- Gurgenashvili et al. (2016) Gurgenashvili, E., Zaqarashvili, T. V., Kukhianidze, et al., 2016, ApJ, 826, 55
- Gurgenashvili et al. (2017) Gurgenashvili, E., Zaqarashvili, T. V., Kukhianidze, et al., 2017, ApJ, 845, 137
- Hathaway (2010) Hathaway, D. H., 2010, Living Rev. Solar Phys. 7, 1 URL: http://solarphysics.livingreviews.org/Articles/lrsp-2010-1
- Heifetz et al. (2015) Heifetz, E., Mak, J., Nycander, J. and Umurhan, O. M., 2015, J. Fluid Mech., 767, 199
- Heng and Spitkovsky (2009) Heng, K. and Spitkovsky, A., 2009, ApJ, 703, 1819
- Heng and Workman (2014) Heng, K. and Workman, J.,, 2014, ApJS, 213, 27
- Howe et al. (2000) Howe, R., Christensen-Dalsgaard, J., Hill, F., et al., 2000, Science, 287, 2456
- Klimachkov and Petrosyan (2017) Klimachkov, D. A. and Petrosyan, A. S., 2017, Phys. Let. A, 381, 106
- Lanza et al. (2009) Lanza, A. F., Pagano, I., Leto, G. et al., 2009, å, 493, 193
- Longuet-Higgins (1968) Longuet-Higgins, M. S., 1968, Proc. R. Soc. London. A., 262, 511
- Lovelace and Hohlfeld (1978) Lovelace, R. V. E. and Hohlfeld, R. G., 1978, ApJ, 221, 51
- Lovelace et al. (1999) Lovelace, R. V. E., Li, H., Colgate, S. A. and Nelson, A. F., 1999, ApJ, 513, 805
- Lou (2000) Lou, Y.Q. 2000, ApJ, 540, 1102
- Lou (2001) Lou, Y. Q. 2001, ApJ, 563, L147
- Matsuno (1966) Matsuno, T.,1966, J. Meteorol. Soc. Japan, 44, 25
- McIntosh et al. (2015) McIntosh, S. W., Leamon, R.J., Krista, L.D. et al., 2015, Nature Communications, 6, 6491
- McIntosh et al. (2017) McIntosh, S. W., Cramer W. J., Marcano M. P., and Leamon, R. J., 2017, Nature Astronomy, 1, 0086
- Oliver et al. (1998) Oliver, R., Ballester, J. L., & Boudin, F. 1998, Nature, 394, 552
- Petviashvili (1980) Petviashvili, V. I., 1980, JETP Letters, 32, 619
- Rieger et al. (1984) Rieger, E., Share, G. H., Forrest, D. J., Kanbach, G., Reppin, C., et al. 1984, Nature, 312, 623
- Rossby (1939) Rossby, C.-G., 1939, J. Marine Research, 2, 38
- Sakurai (1981) Sakurai, K., 1981, Sol. Phys., 74, 35
- Schecter et al. (2001) Schecter, D. A., Boyd, J. F. and Gilman, P. A., 2001, ApJ, 551, L185
- Schwabe (1844) Schwabe, H., 1844, Astron. Nachr. 21, 233
- Spiegel and Zahn (1992) Spiegel, E. A. and Zahn, J.-P., 1992, å, 265, 106
- Umurhan (2010) Umurhan, O. M., 2010, å, 521, A25
- Vecchio et al. (2010) Vecchio, A., Laurenza, M., Carbone, V. & Storini, M., 2010, ApJ, 709, L1
- Zeitlin (2013) Zeitlin, V., 2013, Nonlinear Processes in Geophysics, 20, 893
- Zaqarashvili et al. (2007) Zaqarashvili, T. V., Oliver, R., Ballester, J. L., & Shergelashvili, B. M. 2007, A&A, 470, 815
- Zaqarashvili et al. (2009) Zaqarashvili, T. V., Oliver, R., & Ballester, J. L. 2009, ApJ, 691, L41
- Zaqarashvili et al. (2010a) Zaqarashvili, T. V., Carbonell, M., Oliver, R., & Ballester, J. L. 2010a, ApJ, 709, 749
- Zaqarashvili et al. (2010b) Zaqarashvili, T. V., Carbonell, M., Oliver, R., & Ballester, J. L. 2010b, ApJ, 724, L95
- Zaqarashvili et al. (2015) Zaqarashvili, T. V., Oliver, R., Hanslmeier, A., Carbonell, M., Ballester, J. L., et al. 2015, ApJ, 805, L14