The Dynamics of Magnetic Rossby Waves in the Quasigeostrophic Shallow Water Magnetohydrodynamic Theory

The dynamics of magnetic Rossby waves are investigated by applying a quasigeostrophic shallow water magnetohydrodynamic system, which is linearized with respect to both uniform background flow and uniform magnetic field. Due to the influence of the free surface divergence, the phase speed for magnetic Rossby waves can be either a monotonically increasing or a monotonically decreasing function, and the resulting difference between the group velocity and the phase speed can be either positive or negative. This is determined by whether the corresponding Alfvén wave speed is the upper limit or not. Differently, the phase speed is always a monotonically increasing function and the difference between the group velocity and the phase speed is always positive for incompressible magnetic Rossby waves. Multiplying a factor, the wavenumber vector shares the same endpoint with the group velocity vector. The endpoint moves on a cycle that has a center at the k-axis and is tangent to the l-axis in the wavenumber space. The circle is quite similar to the Longuet-Higgins circle for Rossby waves on Earth’s atmosphere and ocean. The fundamental dynamics is the theoretical basis for deeply understanding the meridional energy transport by waves and the interaction between waves and the background states.


Introduction
Magnetic Rossby waves are widely present in different astrophysical objects such as the solar atmosphere and interior, astrophysical disks, rapidly rotating stars, and planetary and exoplanetary atmospheres (Zaqarashvili et al. 2021).Hide (1966) first studied the dynamics of magnetic Rossby waves in the Earth's core by applying two-dimensional incompressible magnetohydrodynamic (MHD) equations.Considering the fact that the solar tachocline satisfies the conditions for the classic shallow water hydrodynamic (SWHD) equations, Gilman (2000) first established the shallow water MHD (SWMHD) equations and argued that the SWMHD analog may be of use in rotating systems such as the global dynamics of the solar tachocline and the accretion disks.Schecter et al. (2001) solved the analytic solution of the linearized equations with respect to the uniform basic states and pointed out that there are two branches of waves: Alfvén waves and magnetogravity waves if the Coriolis force is absent.Zaqarashvili et al. (2007) further suggested that the two branches of waves become Alfvén and Rossby waves (or magnetic Rossby waves) and magnetogravity waves if the Coriolis force is present.Dellar (2003) further constructed dispersive SWMHD equations that can be applied to investigate the smooth solitary waves and cnoidal wave trains.Dikpati & Gilman (2001) and Gilman & Dikpati (2002) examined the global, hydrodynamic stability of solar latitudinal differential rotation and toroidal field in the solar tachocline using the SWMHD model.
Due to the divergence of horizontal motion, the SWMHD system contains a fast magnetogravity wave component.
However, the influence of the magnetogravity waves can be ignored when considering the slow dynamics at large scale (e.g., Schecter et al. 2001).This means that the fourth-order dispersion relation can be simplified to a second-order one to only retain the low-frequency component: magnetic Rossby waves.This can be done by just comparing the relative sizes of terms in the dispersion relation.However, it works only for static background flow.Derivation becomes complex and corresponding analytic dispersion relation cannot be obtained even for a uniform background flow.In addition, it is inconvenient to understand the basic features of large-scale motion.
To solve this problem, a common practice in the SWHD system is the classic quasigeostrophic (QG) approximation (Charney 1990), which can also be extended to the SWMHD system (Zeitlin 2013).Although the QG forms are similar, the parameters are very different (Balk 2014).The QG equations can also be derived from the rotational compressible MHD flow with well-prepared initial data (Kwon et al. 2018) and from separating the baroclinic modes of the three-dimensional Boussinesq fluid (Balk 2022).Linearization of the QG SWMHD system over the rest state with a constant background magnetic field results in the dispersion relation for magnetic Rossby waves (Zeitlin 2013), which embodies the influence of the divergence of a free surface, and hence slightly different from the one in the two-dimensional incompressible MHD system (e.g., Hide 1966;Zaqarashvili et al. 2007;Li 2022).
Applying the derived dispersion relation, the dynamics of magnetic Rossby waves can be discussed.As a type of dispersive wave, the energy of the magnetic Rossby wave train propagates with the group velocity, which is different from the propagation of the wave phase.The corresponding meridional momentum and energy transport can affect the large-scale dynamics (Dikpati & McIntosh 2020).Therefore, the phase and energy propagation features consist of the fundamental dynamics of magnetic Rossby waves and are the theoretical basis for further studying the interaction between magnetic Rossby waves and the background flow and magnetic field.For example, the longitudinally propagating magnetic Rossby waves in the solar tachocline are closely associated with the surface manifestations in the form of similar propagating coronal holes and patterns of bright points (Dikpati et al. 2018).Magnetic Rossby waves riding on a mean zonal flow may account for some of the geomagnetic westward drifts and have the potential to allow the toroidal field strength within the planetary fluid core (Hori et al. 2018).
There has been some research focusing on this fundamental issue.For example, the relation between the wave energy and wave phase has been preliminarily analyzed in the twodimensional incompressible MHD system (Li 2022).However, there still lacks a systematic and comprehensive investigation, to the best of our knowledge.For Rossby waves on the Earth's atmosphere and ocean, the difference between the zonal group velocity and the zonal phase speed can be used to interpret the observed upstream and downstream effects (Yeh 1949).The relation between the group velocity vector and the wavenumber vector, known as the Longuet-Higgins (LH) circle (Longuet-Higgins 1964), manifests a two-dimensional energy propagation direction in the uniform background flow.The energy dispersion paths called wave rays are a curve in nonuniform background flow, which may interpret the observed atmospheric teleconnection phenomena (Hoskins & Karoly 1981).The meridional energy transport by Rossby waves is fundamental to understanding the interaction between Rossby waves and the Earth's atmospheric west-east background flow and the corresponding meridional circulation.Although there does exist a discrepancy between magnetic Rossby waves and Rossby waves, there do exist similar dynamic features.Therefore, inspired by these classic works, this paper systematically investigated the propagation features and the energy dispersion of magnetic Rossby waves in the QG SWMHD system in uniform background states.

QG SWMHD Equations
The SWMHD equations proposed by Gilman (2000) in the rotating coordinate system can be written as (Schecter et al. 2001;Zaqarashvili et al. 2007) where ( ) u v , and ( ) a b , are components of the horizontal velocity and magnetic field, f is the Coriolis parameter, f = gh is the geopotential height, h is the elevation of the free surface, g is the gravitational acceleration, and d/dt = ∂/∂t + u∂/∂x + v∂/∂y is the individual derivative.Equations (1) and (2) are the horizontal momentum equations; Equation (3) is the mass continuity equation; Equations (4) and (5) are horizontal magnetic induction equations; Equation (6) is the divergence-free condition for the magnetic field (Gilman 2000).
By specifying ( where L and f 0 are the typical scale of the horizontal scale and Coriolis parameter, U and B are typical averaged velocity and magnetic field, the nondimensional forms of Equations (1)-( 6) become where Ro = U/f 0 L is the Rossby number, and λ = B/U is the ratio between the typical magnetic field and velocity.λ can be set to 1 if supposing that the typical values of both magnetic field and velocity are the same (Zeitlin 2013).Here in this paper, it is retained as a parameter.μ 0 = L/L 0 is the ratio between the horizontal scale and Rossby radius of the deformation, which is defined as is the speed of the gravity wave, and H is the averaged thickness of the fluid.The nondimensional Coriolis parameter is where β is the Rossby parameter and is larger than zero.
According to Equation (7) or (8), the magnetic Lorentz force has the same order as the acceleration.This means that the magnetic induction Equations (10) and (11) have the same order as the acceleration terms in Equation ( 7) or (8).In other words, the magnetic induction equations have an order of Ro that is a small parameter for large-scale motion.For convenience, the subscript "1" in Equations ( 7)-( 12) is omitted below.Now expanding u, v, f, a, b as the Taylor series, that is, and taking the zero-order approximation, it is easy to derive the geostrophic wind relations Taking the first-order approximation, it is easy to derive the evolution equations of geostrophic wind and magnetic field In the first-order approximation, the horizontal wind appearing in every term except for the divergence term is replaced by the geostrophic wind.Therefore, the influence of the divergence can be considered.Magnetic induction equations begin to appear in the system and the divergence-free condition for the magnetic field has the simplest form.
According to Equations ( 17)-( 19), it is easy to derive the QG potential vorticity (PV) equation 2 0 is the relative QG PV, and ( ) V B 0 is the vorticity of the magnetic field.Compared with the two-dimensional incompressible MHD system (e.g., Hide 1966), only one extra term ( This manifests the influence of the free surface divergence.Different from the QG PV that is conserved in the SWHD system, the QG PV in the SWMHD system is not conserved and its variation is determined by the magnetic vorticity, the equation of which can be easily derived from Equations (20) and (21).It is It looks a little complex.However, the sum of the last two terms at the right-hand side of Equation ( 24) is zero so that it can become a much simpler form as Simple proof is listed below.According to the nondivergent magnetic field (Equation ( 22)), the magnetic line function ( ) c 0 can be introduced as Since both the magnetic field and geostrophic wind are nondivergent, the geopotential height ( ) f 0 and magnetic line function ( ) c 0 satisfy the relation where F is an arbitrary function.Particularly, if ( ) where C is a constant, the velocity and magnetic field as measured relative to the rotating frame are aligned and everywhere parallel so that the motion can reduce to the corresponding HD one (Acheson & Hide 1973).Substituting Equation (27) into the last two terms in Equation (24), it is easy to obtain the above conclusion.
Applying the magnetic line function, Equation (25) can also be expressed to a simpler form of is the Jacobian operator for any two differential functions m and n.Equation (28) denotes that the magnetic line function is conserved or frozen.Equations ( 23) and ( 25) or (28) consist of the QG SWMHD equations.They keep the most dominant feature for low-frequency large-scale motion.The system also satisfies the energy conservation laws and has its corresponding invariants (e.g., Raphaldini et al. 2023).

Linear Magnetic Rossby Waves
The QG SWMHD system (Equations ( 23) and (28)) filters out the fast magnetogravity waves, which can be regarded as noise for large-scale motion and retains the slow magnetic Rossby waves.The system is linearized with respect to a zonally uniform background flow and a zonally uniform magnetic field, that is, where ū and ā are constants.Then the linearized equations are where k, l are the zonal and meridional wavenumbers and ω is the wave frequency, it is easy to derive the dispersion relation for the magnetic Rossby waves, is the total wavenumber.Equation (32) has two roots, . Note that here the wavenumber k has been set to be positive.Equation (33) means that the magnetic field splits Rossby waves into two modes.The dispersion relation that has the positive (negative) square root corresponds to the fast (slow) mode.And the intrinsic frequency for the fast (slow) mode is always larger (smaller) than zero, namely, w ¢ > 0 f and w ¢ < 0 s , where the subscripts "f" and "s" denote the fast and slow modes, respectively.Equation (33) also demonstrates that there would be no stationary waves for both fast and slow modes if the background flow is static ( ¯= u 0) in which the intrinsic frequency equals the frequency.Equation (33) will return to the dispersion relation for magnetic Rossby waves in the twodimensional incompressible MHD equations if μ 0 = 0 and return to HD Rossby waves if ¯= a 0 (no magnetic field).Equation (33) can be written in the form of the intrinsic zonal phase speed, that is, , where c x = ω/k is the zonal phase speed.The wave will propagate eastward if . Correspondingly, the meridional phase speed is According to Equation (35), for leading structure waves (kl > 0), if they propagate eastward (c x > 0) they will also propagate northward (c y > 0), and for trailing structure waves (kl < 0), if they propagate eastward (c x > 0) they will also propagate southward (c y < 0).Since k has been prescribed to be positive, the leading or trailing structure is determined by the meridional wavenumber l.
The group velocity is defined as where * is the derivative of the intrinsic frequency with respect to the wavenumber.Due to the symmetry of k and l in the expression of intrinsic phase speed, the derivatives are the same.The slope R is a crucial parameter.It can affect the variation of both the phase speed and the group velocity.

Zonal Propagation Feature
For the fast mode, ¢ > c 0 2 2 2 * will be always larger than 0, and hence R f > 0. It means the intrinsic zonal phase speed ¢ c x f , (or the zonal phase speed c x,f ) will be a monotonically increasing function of k.When k → 0, ¢  c 0 x f , is the lower limit; and when k → ∞ , l ¢  c a x f , is the upper limit (Figure 1 , is a monotonically increasing function; and if la is the lower limit ), R f < 0, and , the upper limit is ¯b m -u 0 2 , while the lower limit is la.Correspondingly, ¢ c x f , is monotonically decreasing.Particularly, if ¯= u 0, that is, the static background flow, β * = β > 0. The lower limit will be zero and the upper limit will be la, which corresponds to the monotonically increasing case, and hence , the lower limit equals the upper one, and hence R f = 0, that is, c x,f is independent of the zonal wavenumber k and the waves will become nondispersive.If m = 0 0 2 , β * = β > 0 and R f > 0. Therefore, ¢ c x f , will be always monotonically increasing for fast incompressible magnetic Rossby waves where the effect of the divergence is neglected.According to the above analysis, the background information can strongly affect the propagation of the fast mode, especially when the zonal wavenumber is relatively small or the horizontal scale of the wave is relatively large.The influence weakens when the horizontal scale of the wave shrinks to be small scale.And the fast mode becomes closer and closer to the corresponding fast Alfvén wave.
For the slow mode, ¢ < c 0 , is a monotonically decreasing function due to −R s < 0, and its upper limit is zero when k → 0, and its lower limit is Similarly, it is easy to prove (see Appendix B) that if l a is the the two limits will be equal, and hence R s = 0, which means ¢ c x s , is independent of k and the waves will become nondispersive.And if . Furthermore, the limit b m - -¥ 0 2 will be always the lower limit.Therefore, ¢ c x s , will be always a monotonically increasing function for slow incompressible magnetic Rossby waves.Similar to the fast mode, the background information also manifests its influence on the large-scale motion.
The introduction of the background flow makes it possible for stationary waves (c x = 0) to exist.For the fast mode, when β * > 0 (equivalent to ¯b m >u 0 2 ), the phase speed monotonically varies from ū to ¯l + u a.To make sure there is a zeropoint for the phase speed, this requires ¯< u 0 and ¯l + > u a 0. To combine these conditions together, we get ), the phase speed monotoni- , it is obvious that there is no possibility for c x,f = 0.If ¯l + u a is the lower limit, then ¯l + < u a 0 is necessary to make sure c x,f = 0. Combining these conditions together, we get For both cases, the fast stationary magnetic Rossby waves cannot propagate in a westerly background.This feature is totally opposite to Rossby waves, which cannot propagate in an easterly background.
For the slow mode, when ), the phase speed varies from ¯l - u a to ū.This requires ¯l -< u a 0 and ¯> u 0. It is obvious that there are no stationary waves.When , which means no stationary waves.If ¯lu a is the upper limit, then ¯l - > u a 0 is necessary to ).We know that the group velocity denotes the propagation speed of the wave energy and the phase speed denotes the phase propagation speed.Therefore, a faster group velocity in the same direction as the phase speed means that the wave energy can propagate ahead of the phase propagation to arrive downstream to induce new perturbations.This is the upstream effect.Correspondingly, a slower group velocity in the same direction means that the wave energy propagates behind the wave phase to induce new perturbations or enhance the existing perturbations upstream.This is the downstream effect.
For the static background flow ( ¯= u 0), there is only an upstream effect for the fast mode since c gx,f > c x,f > 0 and there is only a downstream effect for the slow mode since c gx,s > c x,s and c x,s < 0. This is consistent with the previous investigation of Li (2022).The upstream and downstream effects are well known for Rossby waves on the Earth's atmosphere and ocean.However, they are rarely discussed for magnetic Rossby waves.

Zonal and Meridional Propagation Feature
The above analysis mainly focuses on the zonal propagation features for magnetic Rossby waves.Now we will discuss the two-dimensional propagation features.Equations ( 36) and (37) can be written to or to a vector form of where ¢ c g can be called relative group velocity vector.Equation (42) means the relative group velocity vector is parallel to the wavenumber vector.The group velocity vector can be written as x Equation (43) means that the group velocity vector will share the same endpoint but with the different starting point with the relative group velocity or the wavenumber vector multiplied a factor Rk in the wavenumber space.For the relative group velocity vector, the starting point is located on the coordinate origin.For the group velocity vector, the starting point (P) has an increment of ±c x /Rk in the k direction (Figure 4).Note that the sign of the increment is determined by the signs of c x /R.If c x /R > 0, the increment is located on the negative (positive) half of the k direction for the fast (slow) mode, which propagates energy northeastward (southwestward) as Figure 4 denotes.The corresponding meridional energy transport can affect the background flow and magnetic field there.Fixing the total wavenumber K, Equation (42) can be rewritten as in the polar coordinate (ρ, θ), where ρ is the length of the relative group velocity vector and θ is the angle between the relative group velocity vector and the k-axis.It is obvious that Equation (44) manifests that the trajectory for the endpoint (W) of the relative group velocity vector will be a cycle with a center at point Q (±RK 2 /2, 0) in the k-axis and a radius of in the wavenumber space (Figure 4).It is interesting to point out that this cycle is quite similar to the classic LH circle (Longuet-Higgins 1964) on the Earth's atmosphere, which states that the starting point of the wavenumber vector is located on the coordinate origin and the endpoint is located on the cycle that has a center of (−β/2ω, 0) and a radius of 0 2 in the wavenumber space.Equation (42) or (44) can be called the LH circle for magnetic Rossby waves.Note that there are differences between them.First, the LH circle for Rossby waves is derived by fixing the frequency, while the LH circle for magnetic Rossby waves is derived by fixing the total wavenumber.Second, the LH circle for Rossby waves deals with the wavenumber vector, while the LH circle for magnetic Rossby waves deals with the relative group velocity vector, which is parallel to the wavenumber vector but with a different length.Finally, the LH circle for Rossby waves is tangent to the l-axis if divergence is neglected, while it has a distance from the l-axis if divergence is present.The LH circle for magnetic Rossby waves is always tangent to the l-axis no matter whether divergence is considered or not.Of course, the parameter R is determined by the divergence of the free surface.These formulas had been first derived by Li (2022).However, Li did not discuss the LH cycle conclusion comprehensively.Here are the latest updates.
According to dispersion relation Equation (32), it is also easy to derive a similar formula by fixing the frequency formally describes a cycle that has a center point (−ηβ * /2ω, 0) and a radius of R 1 in the wavenumber space.However, since η (ω A ) also varies with the zonal wavenumber, the trajectory will no longer be a cycle in the wavenumber space.If neglecting the influence of the Alfvén waves (ω A = 0), Equation (45) will reduce to the classic LH circle for Rossby waves.

Conclusions and Discussion
The dynamics of the magnetic Rossby waves are investigated in this paper by linearizing the QG SWMHD system with respect to the uniform background flow and magnetic field.Although the dispersion relation is similar to magnetic Rossby waves in the incompressible MHD system, the dynamics present significant discrepancies.For the incompressible magnetic Rossby waves, the zonal phase speed is always a monotonically increasing function of the zonal wavenumber from a lower limit that equals the background flow (that tends to be negative infinity) to an upper limit that equals the corresponding fast (slow) Alfvén wave speed.For magnetic Rossby waves in the QG SWMHD system, the propagation becomes more complex.There exists a totally different case in which the corresponding fast or slow Alfvén wave speed becomes the lower limit due to the influence of the background flow.The zonal phase speed becomes a monotonically decreasing function in this case.In addition, the difference between the zonal group velocity and the zonal phase speed becomes negative in this case, which is also quite different from the incompressible one, in which the difference is always larger than zero for both modes.
The group velocity vector shares the same endpoint with the wavenumber vector multiplying a factor, called the relative group velocity vector but with different starting points.The relative group velocity vector is parallel to the wavenumber vector and thereby the starting point located on the coordinate origin in the wavenumber space.The starting point of the group velocity, however, has an increment in the k-axis.If the horizontal scale of waves is specified, the trajectory of the endpoint will be a circle the center of which is also located in the k-axis.The circle is also tangent to the l-axis.This circle is quite similar to the LH circle for Rossby waves on the Earth's atmosphere and ocean.Besides, the circle is formally the same for both incompressible and QG systems but with different centers and radii.
As Zeitlin (2013) remarked, the QG approximation might help in establishing a common language with the astrophysical and geophysical fluid communities.These results also generalize the standard analyzing method in the earth's atmosphere and ocean.The present work can also be extended to zonally varying background flow and magnetic field, in which the energy dispersion trajectories are curves due to the influence of the varying background states.As Li (2023) suggested, there would exist some unique characteristics for the magnetic Rossby wave rays in the incompressible MHD system.It can be expected that these new features can also be observed in the QG SWMHD system.This deserves further investigation. .This means that R f > 0 always holds.Therefore, we mainly focus on the case when β * < 0 (equivalent to ¯b m < -< u 0 0 2 ).The two limits become ¯b m -u 0 2 when k → 0 and la when k → ∞.
, then la is the upper limit, that is, , then la is the lower limit, that is, which is also contradictory to the condition.Therefore, Equation (A7) cannot hold and its opposite R f < 0 is correct.

Figure 1 .
Figure 1.The variation of the zonal intrinsic phase speed ¢ c x f , with the zonal wavenumber k.(a) For the case where β * > 0 and (b) for β * < 0.
R s < 0 and thereby a monotonically increasing ¢ c x s , (Figure 2(b)).To summarize, when ¯l b m < u a 0 2 , R s > 0. The upper limit is either zero (when ¯) b m < -, if ¯= u 0 for the static background flow, β * = β > 0. The variation of ¢ c x s , depends on the relative sizes of the two limits, l a and b m -

Figure 2 .
Figure 2. The variation of the zonal intrinsic phase speed ¢ c x s , with the zonal wavenumber k.(a) is for the case where β * < 0 and (b) is for β * > 0.

Figure 3 .
Figure 3.The variation of the difference between the zonal group velocity and the zonal phase speed with the zonal wavenumber k.(a) is for the fast mode and (b) is for the slow mode.

Figure 4 .
Figure 4. LH circle for magnetic Rossby waves.(a) is for the case where R f > 0 or R s < 0; (b) is for the case where R f < 0 or R s > 0.
to the condition.Therefore, Equation (A3) does not hold and its opposite R f > 0 is correct.