The energy dispersion of magnetic Rossby waves in zonally nonuniform basic states

The energy dispersion of magnetic Rossby waves was investigated by applying two-dimensional incompressible magnetohydrodynamic equations in both zonally varying basic flow and basic magnetic field. A derived cubic dispersion relation suggests that there are at most three types of magnetic Rossby wave. Two of them represent waves that gradually tend to Alfvén waves during the energy dispersion process. The energy dispersion trajectories (wave rays) finally move with the zonal group velocity that tends to be equal to the zonal phase speed after being reflected by at least one turning location at which the meridional group velocity equal to zero. Along the marching rays, both the wave action density and wave energy tend to be constant values while the wave amplitude will decrease with increasing total wavenumber. The third one represents a wave that gradually have the constant meridional wavenumber, wave action density, wave energy, and wave amplitude. However, the difference in the zonal group velocity and the zonal phase speed suggests that the wave is still dispersive. This type of wave will disappear if specifying uniform basic magnetic field. The cubic dispersion relation is then reduced to a quadratic one. Correspondingly, the remaining two dispersion relations feature a fast- and a slow-propagating magnetic Rossby wave, respectively. They finally tend to be Alfvén waves with no energy dispersion when the energy dispersion process completes.


Introduction
Rossby waves on the Earth's atmosphere are the basis for understanding the weather change and short-term climate variability.Since Rossby (1939) firstly described them physically, the theory of Rossby waves has been constructed comprehensively.As for the energy dispersion of Rossby waves, Yeh (1949) firstly discussed the discrepancy between the zonal group velocity and phase speed, interpreting the observed downstream and upstream phenomena.Longuet-Higgins (1964) further discussed the two-dimensional energy dispersion.He pointed out that endpoints of the group velocity vector and the wave vector are on a cycle on the wavenumber space.Hoskins and Karoly (1981) further discussed the influence of the basic flow.They proved that the energy dispersion paths (called wave rays) are great cycles on sphere if specifying a super rotating basic flow.Wave ray theory can be applied in interpreting teleconnection phenomena on the Earth's atmosphere and thereby has been extensively investigated (Karoly and Hoskins 1982, Karoly 1983, Hoskins and Ambrizzi 1993, Li and Nathan 1994, Yang and Hoskins 1996, Li and Nathan 1997, Li et al 2015).Recently, Li et al (2021a) derived a formula to solve the divergence of group velocity and discussed the variations in wave energy and amplitude (Li et al 2021a, Li et al 2021b, 2022, Li and Kang 2022).
These investigations benefit in-depth understanding for the energy dispersion of Rossby waves, which are described based on hydrodynamic (HD) equations.In circumstances the magnetic field cannot be ignored, Rossby waves become to magnetic Rossby waves that are described by the magnetohydrodynamic (MHD) equations.Magnetic Rossby waves play an important role in the large-scale dynamics of different astrophysical objects such as the solar atmosphere and interior, astrophysical discs, rapidly rotating stars, planetary and exoplanetary atmospheres (Zaqarashvili et al 2021).For example, Magnetic Rossby waves were firstly analyzed and applied to interpret the observed properties in the Earth's core (Hide 1966).Following the same analysis method as Rossby waves in the Earth's atmosphere and ocean, the basic dynamics of magnetic Rossby waves, such as propagation, instability, have been further discussed (Gilman 1969, Acheson andHide 1973).Coupled Rossby-Alfvén-Khantadze electromagnetic planetary waves can exist in the Earth's ionosphere E-layer (Kaladze et al 2004).They also can propagate in the solar tachocline (Leprovost andKim 2007, Zaqarashvili et al 2007) in MHD shallow water equations that are proposed by Gilman (2000).The phase propagation of magnetic Rossby waves in the solar tachocline, a typical characteristic of tachocline nonlinear oscillations are also responsible for producing solar seasons (Dikpati et al 2017(Dikpati et al , 2018a(Dikpati et al , 2018b)).
Previous advances highlighted the dynamics of magnetic Rossby waves but leave the energy dispersion undiscussed.It is easy to derive the group velocity from the dispersion relation (e.g., Hide 1966, Zaqarashvili et al 2021).And if the basic magnetic field and motion field are uniform, the group velocity components are constant values and the rays propagate in straight lines (Acheson and Hide 1973).Recently, Li (2022) discussed the upstream and downstream effects for magnetic Rossby waves and pointed out the endpoints of group velocity vector and wave vector are on a cycle on wavenumber space, analogical to the ones for HD Rossby waves.However, the energy dispersion paths (called rays) are straight lines due to uniform basic states.Besides, there will be no variation in wave energy and amplitude along the rays.Considering the situation will be quite different if introducing the zonally varying basic states, I further extend the energy dispersion to the zonally varying basic states.First, I derive the dispersion relation of magnetic Rossby waves in the barotropic incompressible MHD equations.Then the wave rays are calculated according to the group velocity.Finally, I explicitly discuss the dynamics and characteristics of the energy dispersion.

The wave ray theory
The two-dimensional incompressible MHD equations in Cartesian geometry are where u v , ( ) and a b , ( ) are fluid velocity component, and magnetic field components in the x and y directions, respectively; p is pressure; b x a y B V = ¶ ¶ - ¶ ¶ is the vertical vorticity of the magnetic field; f f y is the Coriolis parameter and f 0 is its value at a reference location; b is the slope of f and is taken to be a constant value.It is the commonly applied bplane approximation in geophysics.The nondivergent conditions for motion field and magnetic field are u x v y 0 ¶ ¶ + ¶ ¶ = and a x b y 0, ¶ ¶ + ¶ ¶ = respectively.I note that although the equations may be some repetition with the previous paper (Li 2022), I still retain them for the sake of the completeness of this investigation.
The vertical vorticity of velocity field can be derived from equations (1) and (2).It is The vertical vorticity of magnetic field can be derived as follows.Let's first differentiate equation (3) with respect to y to obtain Now let's subtract equation (6) from equation (7) to derive the vertical vorticity equation for magnetic field.It is According to Alfvén's theorem that states in a perfectly conducting fluid (e.g., as equations (3) and (4) denote) magnetic field lines move with the fluid without dissipation, one can write where y and c are streamfunction and magnetic field line function, respectively.They are introduced by the nondivergent motion and magnetic field so that a y , . 2  = ¶ ¶ + ¶ ¶ is the horizontal Laplace operator.Here I note that m represents an arbitrary function.Applying equation (9), the relations between velocity and magnetic field are where m c ¢( ) is the derivative of m with respect to .c With the help of equations ( 10) and (11), the last two terms of equation (8) become so that it can be simplified to a simple form, that is Linearizing equations (13) and (14) with respect to the zonally varying basic flow and basic magnetic field in present investigation, namely where u y ¯( ) and a y ¯( ) are the zonally varying basic flow and basic magnetic field, respectively, I can derive ¯is the gradient of absolute vorticity with respect to y; y¢ and c¢ are perturbations for the streamfunction and magnetic field line function, respectively.
I then introduce the normal mode solution with two-dimensional propagation i kx ly t i kx ly t where k l , are the zonal and meridional wavenumber, respectively.w is the frequency.Y and C are the slowly varying amplitude for streamfunction and magnetic field line function, respectively.
x Y y T t X , , e e e = = = and e is a small parameter that divides the slowly varying amplitude of the wave packet (varying with X Y T , , ) and the relatively fast varying in wave phase (varying with x y t , , ).I also assume that both u ¯and a ¯are slowly varying and the amplitude of the wave packet are also slowly varying comparing to the wave phase.
where uk w w ¢ = -¯is the intrinsic frequency; is the total wavenumber square.It is a cubic algebraic equation of frequency and has three real roots at most.Therefore, there will be at most three types of magnetic Rossby waves.Let us set w ˆ, 2 w ˆand 3 w ˆare three unequal real roots, representing three types of neutral magnetic Rossby waves.I note that there are only two modes of magnetic Rossby waves if the basic states are zonally uniform as in the previous study (Li 2022) w ˆis a real root while 2 w ˆand 3 w ˆare two complex roots, which denote a unstable magnetic Rossby wave that propagates with the real part of the 2 w ˆ(or 3 w ˆsince they are conjugate) and develops with a growth rate of the absolute imagery part of . 2 w ˆIt is obvious that the condition for 0 3 D > may help deriving the sufficient and necessary conditions for magnetic Rossby waves instability.
The group velocity is where i 1, 2, 3 = denotes the three dispersion relations, respectively.Correspondingly, a ray is defined as to describe a spatial curve, the tangent direction of which is the direction of the group velocity vector.According to Bretherton and Garrett (1969), I directly write the wave action density equation where ¢ is the wave action density and is the wave energy density that consist of the kinetic energy and the magnetic energy, where A 0 is the wave amplitude.I note that we can also derive equation (21) by substituting equations (19) and (18) into equations ( 16) and (17), and taking the first-order approximation (namely, Equation (28) can be transformed to along a ray is the derivative along a moving ray, and c g is the group velocity vector.There are also some useful relations which suggest that the frequency and zonal wavenumber are constant values along a moving ray and the variation in the meridional wavenumber is determined by the derivative of the dispersion relation with respect to y. Equation (30) means that the variation in the wave action density along a moving ray is determined by the divergence of group velocity.If the group velocity is divergent (convergent), the wave action density will decline (intensify) along the moving ray.However, the divergence of group velocity cannot be directly calculated according to equation (27).This is because despite the group velocity along a ray is known, the group velocity neighboring a ray is unknown (Lighthill 1978).To solve the wave action density and hence wave energy and amplitude, Li et al (2021a) proposed a method where t denotes time and t d denotes a short time interval; S d denotes the unit area that is perpendicular to the local group velocity vector.Based on the method, the variations in wave energy and amplitude along a moving ray are extensively discussed for Rossby waves on the Earth's atmosphere (Li et al 2021b, 2022, Li and Kang 2022).

Results and discussions
which represent a fast-propagating and a slow-propagating magnetic Rossby wave, respectively.Equations ( 35) and (36) are formally the same as the magnetic Rossby waves in the uniform basic flow (Hide 1966, Acheson and Hide 1973, Gilman 2000, Zaqarashvili et al 2007, Dikpati et al 2020).However, the group velocity will be different since * a 2 ¯varies with the zonal and meridional wavenumber.The group velocity is ) ¯is the discriminant.The last terms in equations (37) and (38) manifest the influence of the zonally varying basic flow.They will disappear if the zonal basic flow is uniform and corresponding group velocity will reduce to the one in the zonally uniform basic flow.
I further calculate one case to exhibit the rays and variations in wave energy and amplitude along the rays.In both two cases, the starting position is set to (0, 0) in x y , ( ) plane (the boundary of the westerly and easterly wind) and the zonal and meridional wavenumbers are set to 1.The fast-propagating magnetic Rossby waves propagate eastward (or prograde) while the slow-propagating magnetic Rossby wave propagate westward (or retrograde).
The ray (figure 1(a)) of the fast-propagating magnetic Rossby wave moves northeast toward a northernmost place and then turns its direction to move southeast toward a southern location when the calculation time ends.Similar to the practice for Rossby waves on Earth's atmosphere, the northernmost place is called a turning location (or turning latitude) where the meridional wavenumber equals zero and the ray is reflected to an opposite direction in the y-direction.The southern location is called a critical location (or critical latitude).The ray will move toward but never arriving at the critical location where the meridional wavenumber square tends to be infinity.Besides, the zonal group velocity will tend to equal to the zonal phase speed when the ray tends to the critical location and the meridional group velocity tends to vanish.This suggests that the wave will finally move eastward with the zonal phase speed and become nondispersive, or the wave will finally tend to the corresponding Alfvén wave after the energy dispersion process completes.
Along the ray, wave action density (solid curve in figure 1(b)) fluctuates when the ray moves near the turning location and tends to a constant value when the ray finally moves toward the critical location.The variation in intrinsic frequency (dashed curve in figure 1(b)) is only determined by the zonal basic flow.It has a minimum value at the turning location and has a maximum value at the critical location.The variation in wave energy density (solid curve in figure 1(c)) is quite similar to that in wave action density.The in-phase variation in wave energy density and action density suggests that the wave energy density plays a dominant role in determining the wave action density.Furtherly, the both kinetic energy (dashed curve in figure 1(c)) and magnetic energy (dash-dotted curve in figure 1(c)) contribute to energy variation near the turning location.When the ray moves toward the critical location, the increase in wave energy is caused by increasing kinetic energy rather than decreasing magnetic energy.The total wavenumber (dashed curve in figure 1(d)) is minimum at the turning location (caused by zero meridional wavenumber) and continuously increases when the ray moves toward the critical location.Therefore, amplitude (solid curve in figure 1(d)) shows opposite variationapproaching maximum value at the turning location and then continuously decreasing.Comparing with the initial values that are set to 1, the variations in both wave energy and amplitude are moderate.It implies that the wave may not develop or decay significantly near the turning location.
The ray (figure 2(a)) of the slow-propagating magnetic Rossby waves shows certain discrepancy.It begins to move northwest to a westernmost place and then move northeast toward the turning location.After that, it moves southeast to an easternmost place and then moves southwest with continuously increasing integration time.Near the turning location, the ray forms a cycle-like structure, which is caused by the local values of group velocity.The ray moves westward (eastward) if the zonal group velocity is smaller (larger) than zero and move southward (northward) if the meridional group velocity is smaller (larger) than zero.
Along the ray, the wave action density (solid curve in figure 2(b)) approaches to the maximum value at the easternmost and the westernmost place and minimum value at the turning location while the intrinsic frequency (dashed curve in figure 2(b)) has the minimum value at the turning location.The wave energy density (solid curve in figure 2(c)) shares the same variation as the wave action density.Also, it is the kinetic energy (dashed curve in figure 2(c)) that mainly contributes to variation in wave energy while the magnetic energy (dash-dotted curve in figure 2(c)) only accounts for a small proportion.Both wave energy and total wavenumber (dashed curve in figure 2(d)) contribute to variation in wave amplitude (solid curve in figure 2(d)) which fluctuate near the cycle-like structure and finally decreases with continuously increasing total wavenumber.The variations in wave energy and amplitude near the cycle-like structure are moderate and are not always in-phase, suggesting that the wave may neither significantly develop nor significantly decay.Also, when the ray continuously moves toward the critical location with continuously increasing total wavenumber (or shrinking spatial scale), the zonal group velocity tends to be zonal phase speed and the meridional group velocity tends to vanish, the energy dispersion speed will be close to the phase speed.The slow-propagating magnetic Rossby wave will also become nondispersive and tend to be close to the corresponding slow-propagating Alfvén wave.
According to above analysis, I may conclude some characteristics of the energy dispersion of magnetic Rossby waves.The first one is that the energy dispersion region may be limited in a region enclosed by a turning location and a critical location.This is similar to Rossby waves on the Earth's atmosphere (e.g., Li et al (2021a)).
In the energy dispersion region, a ray can form a complex cycle-like structure, which is also similar to westward propagating Rossby waves in tropical easterly (e.g., Li and Kang (2022)).The second one is that both wave energy and amplitude approach extreme values at the turning location.The variations in both wave energy and amplitude along a ray that is marching toward (or away from) a turning location can be applied to discuss the development of magnetic Rossby waves.This is also quite similar to Rossby waves.The last one is that the zonal phase speed determines the final direction of a ray.For example, the ray will finally move westward if the wave's zonal phase speed is negative although the ray itself may moves eastward at the beginning time.Also, when the ray moves toward the critical location, the magnetic Rossby waves will tend to be corresponding Alfvén waves with no energy dispersion and the wave energy will tend to be constant values while amplitude decreases with increasing total wavenumber (or shrinking spatial scale).This is different from Rossby waves, both wave energy and amplitude of which will decrease to zero when rays move closer and closer to the critical location.

Both zonally varying basic flow and magnetic field
The dispersion relation equation ( 13) cannot be simplified in cases where both the zonal basic flow and basic magnetic field are nonuniform.I specify a basic magnetic field that has the same variation with the basic flow but with a weaker magnitude, namely The zonal and meridional wavenumbers are also set to 1 and the rays start from (0,1) in x y , ( ) plane.The dispersion relation has three unequal real roots with these specified parameters, denoting three types of waves.The first is a westward propagating one with a negative frequency; the second and the third ones are eastward propagating, with positive frequencies.Here I note that the three unequal real dispersion relations depend on the sign of the discriminant.With above specified parameters values, the discriminant is smaller than zero on almost the entire y coordinate axis except for a little region (figure 3(a)), where there are two complex dispersion relations.It is obvious that magnetic Rossby wave will be unstable in this region.Therefore, it provides a sufficient and necessary condition for instability of magnetic Rossby waves with specified basic states.I further analyze the sign of the discriminant by only fixing one variable.When the meridional wavenumber is set to 1, the region where the discriminant is larger than zero is located in y 0 < part with small zonal wavenumber or large spatial scale (figure 3(b)).The situation is quite similar for the case where the zonal wavenumber is set to 1 (figure 3(c)).If the initial location is set to (0, −1), the region is a ring, meaning that unstable waves require the wavenumber or the spatial scale can neither too large nor too small (figure 3(d)).Generally speaking, the unstable region is located in a range for the zonal wavenumber, meridional wavenumber and the location in y axis.Of course, it obviously depends on the basic states.
Figure 3 The variation in the discriminant with y (a) and the shaded regions where the discriminant is larger than zero (b, c, d).
The ray (figure 4(a)) of the first type of wave shows quite different features.It moves northeastward with basically constant zonal and meridional group velocity to form a near straight line trajectory.Along the ray, the wave action density (solid curve in figure 4(b)), the intrinsic frequency (dashed curve in figure 4(b)), and the wave energy density (solid curve in figure 4(c)) decreases to constant values.Besides, although the kinetic energy (dashed curve in figure 4(c)) dominates the wave energy, the magnetic energy (dash-dotted curve in figure 4(c)) also plays a minor role.The total wavenumber (dashed curve in figure 4(d)) gradually decreases to a constant value (caused by gradually decreasing meridional wavenumber), denoting the spatial scale of the wave will also be unchangeable.This determines the amplitude will also tend to a constant value.I note that when the ray enters regions where y 3, > both basic flow and basic magnetic field are basically uniform, which may cause the nearly constant zonal and meridional group velocities and hence nearly constant wave energy and amplitude.Nevertheless, this type of wave is quite different from the previously discussed fast-propagating and slowpropagating magnetic Rossby wave.The ray (figure 5(a)) of the second type of magnetic Rossby wave moves northeast and then southeast after being reflected by a turning location.The ray will finally move eastward with the positive zonal phase speed with continuously increasing total wavenumber.Therefore, the wave finally tends to be an Alfvén wave with no energy dispersion.Both the wave energy density (solid curve in figure 5(c)) and the amplitude (solid curve in figure 5(d)) approach to the maximum values at the turning latitude.The maximum values are not very large compared to the initial values.It implies that the wave may not significantly develop when the ray moves toward the turning location.
The ray (figure 6(a)) of the third type of magnetic Rossby wave will also finally move eastward with the positive zonal phase speed after being alternatively reflected by two turning locations.This is also similar to Rossby waves on the Earth's atmosphere.A Rossby wave ray can propagate between two turning locations to form a wave-like structure (e.g., Li et al (2021a)).However, the two turning locations in present case are formed differently.The southern one is caused by the zero meridional wavenumber and hence zero meridional group velocity while the northern one is caused by the zero meridional group velocity, rather than the zero meridional wavenumber.This is caused by complex form of meridional group velocity that not only has one zero point at zero meridional wavenumber but also other zero points without zero meridional wavenumber.Therefore, the northern turning location only reflects the ray but without any influence on the wave action density, wave energy density and amplitude.Both wave action density (solid curve in figure 6(b)), wave energy (solid curve in figure 6(c)) and amplitude (solid curve in figure 6(d)) approach to the maximum values, that are significantly intensified compared to their initial values, at the southern turning location.It implies that the wave may significantly develop before when it propagates toward the southern turning location.
Here I conclude some new features in energy dispersion of magnetic Rossby waves.The first one is that if there are three types of waves, one of them are quite different from the other two.It represents a wave that will always keep its energy dispersive feature.The other two waves will finally tend to corresponding Alfvén waves with no energy dispersion.The second one is that there may be two turning locations, one of which is caused by the nonzero meridional wavenumber.Meanwhile, there will also be a critical location.There are also lots of studies to be conducted to identify the conditions of each turning location and the critical location.The last one is that both wave energy and amplitude may have dramatic intensifications at the southern turning location, implying a magnetic Rossby wave may significantly develop when it moves toward the turning location.

Conclusions
This paper investigates the energy dispersion of the magnetic Rossby waves under the zonally varying basic flow and basic magnetic field by applying the linearized two-dimensional incompressible MHD equations.The results show that the dispersion relation is a cubic equation of the frequency.If the discriminant is larger than zero, there will be a real dispersion relation and two complex dispersion relations, which represent the neutral and unstable waves, respectively.If the discriminant is smaller than zero, there will be three real dispersion relations, each of which features a kind of magnetic Rossby wave.The cubic dispersion relation can be simplified to a quadratic one if specifying the uniform basic magnetic field but the zonally varying basic flow.The quadratic dispersion relation is formally same as the one in the both uniform basic flow and basic magnetic field.It denotes a fast-and a slow-propagating magnetic Rossby wave.
I firstly discuss the energy dispersion process by specifying a basic flow that highlights the wind shear from a basically uniform easterly to a basically uniform westerly.The ray of the fast propagating magnetic Rossby wave moves eastward when it starts at the boundary of the easterly and westerly while the corresponding ray for the slow wave propagates westward.Both two waves will finally tend to be the corresponding Alfvén waves with no energy dispersion when the energy dispersion completes.Rays of the fast (slow) waves will finally move eastward (westward) after passing by one turning location.Both the wave energy and amplitude approach to extreme values at the turning location that is caused by zero meridional wavenumber.If both of them have significant increases, the wave may experience significant development before it arrives at the turning location.However, it is still unclear under what circumstances both the wave energy and amplitude can significantly increase at the same time.It deserves further investigation.
I then discuss the energy dispersion process by specifying both zonally varying basic flow and basic magnetic field.There are three real unequal dispersion relations that represent three kinds of waves if the initial position is set to (0, 1) and both the zonal and meridional wavenumbers are set to 1. Two of them, which propagate eastward, follow the same variations.The waves will final become corresponding Alfvén waves with no energy dispersion.The rays will finally move eastward after being reflected by at least one turning location.The wave energy will tend to be constant values while the wave amplitude will decrease with continuously increasing total wavenumber.The left one, which propagate westward, shows quite different features.The ray will move to northeast and finally tend to a straight line with constant positive zonal and meridional group velocities.Besides, the wave action density, wave energy density, wave amplitude all tend to be constant values along the marching ray.The fact that the zonal group velocity is different from the zonal phase speed suggests the wave will be always dispersive.This behavior is quite similar to Rossby waves, rather than the Alfvén waves that are non-dispersive.

I
further apply the Wenzel-Kramers-Brillouin (WKB) approximation to expand Y and C into series of the small parameterSubstituting equations (19) and (18) into equations (16) and (17), and taking the zero-order approximation, namely 0 Y = Y and 0 C = C which means the local amplitude is considered to be constant values since they vary slowly, I derive the dispersion relation 3.1.A zonally varying basic flow and a uniform basic magnetic field Let's begin with a simple case in which the zonal basic flow is set to basic magnetic field is set to uniform.According to previous analysis (Li 2022), the zonal basic magnetic field can neither be too strong nor too weak to feature magnetic Rossby waves.Therefore, a moderate zonal basic magnetic field a 0.5 = ¯is specified.I also tried different values (e.g., a 1.0 = ¯) and there is no significant difference.Equation (26) suggests that when y 3large enough, the basic flow also tends to be uniform.The dispersion relation equation (20) can be reduced to a quadratic one by just discarding the terms with the derivatives of the basic magnetic field.It is ¯¯¯is the correction of the basic flow by the basic magnetic field.It is easy to solve it two roots

Figure 1 .
Figure1.Ray trajectory (solid black dots denoting the unit time interval) for the fast-propagating magnetic Rossby wave that starts from (0, 0) with zonal and meridional wavenumbers equal to 1 (a); variations in wave action density (solid curve) and intrinsic frequency (dashed curve) along the ray trajectory (b); variations in wave energy density, kinetic and magnetic energy along the ray trajectory (c); and variations in wave amplitude and total wavenumber along the ray (d).

Figure 2 .
Figure 2. Same as figure 1 but for the slow-propagating magnetic Rossby wave.

Figure 4 .
Figure4.Ray trajectory (solid black dots denoting the unit time interval) for the first type of wave that starts from (0, 1) with zonal and meridional wavenumbers equal 1 (a); variations in wave action density (solid curve) and intrinsic frequency (dashed curve) along the ray trajectory (b); variations in wave energy density, kinetic and magnetic energy along the ray trajectory (c); and variations in wave amplitude and total wavenumber along the ray (d).

Figure 5 .
Figure 5. Same as figure 3, but for the second type of magnetic Rossby wave.