Nonlin. Processes Geophys., 20, 893–898, 2013 Open Access www.nonlin-processes-geophys.net/20/893/2013/ Nonlinear Processes doi:10.5194/npg-20-893-2013 © Author(s) 2013. CC Attribution 3.0 License. in Geophysics

Remarks on rotating shallow-water magnetohydrodynamics

V. Zeitlin Institut Universitaire de France, Laboratoire de Météorologie Dynamique, UMR8539, CNRS – University P. and M. Curie and Ecole Normale Supérieure, Paris, France

Correspondence to: V. Zeitlin ([email protected])

Received: 8 March 2013 – Revised: 19 July 2013 – Accepted: 25 September 2013 – Published: 29 October 2013

Abstract. We show how the rotating shallow-water MHD done in geophysical applications of RSW (cf., e.g., Pedlosky, model, which was proposed in the solar tachocline context, 1982; Zeitlin, 2007). Shallow-water and QG approaches be- may be systematically derived by vertical averaging of the ing standard working tools in the atmosphere–ocean commu- full MHD equations for the rotating magneto fluid under the nity, the present work might help in establishing a common influence of gravity. The procedure highlights the main ap- language with the astrophysical community. proximations and the domain of validity of the model, and allows for multi-layer generalizations and, hence, inclusion of the baroclinic effects. A quasi-geostrophic version of the 2 Derivation of the mRSW equations model, both in barotropic and in baroclinic cases, is derived 2.1 Vertical averaging of the MHD equations in the limit of strong rotation. The basic properties of the model(s) are sketched, including the stabilizing role of mag- Our starting point is the system of three-dimensional com- netic fields in the baroclinic version. pressible MHD equations on the rotating plane in the pres- ence of gravity: 1 ∂ v + v · ∇v − b · ∇b + gzˆ + f zˆ ∧ v + ∇p∗ = 0, (1) 1 Introduction t ρ + · ∇ − · ∇ + ∇ · = The (rotating) shallow-water magnetohydrodynamics model ∂t b v b b v b v 0, (2) was introduced on heuristic grounds in Gilman (2000) in the ∂t ρ + ∇ · (ρv) = 0, ∇ · b = 0, (3) context of the solar tachocline. (We will call it mRSW in = = what follows; with respect to the originally used acronym where v (v1,v2,v3) is the fluid velocity, b (b1,b2,b3) is sMHD, this one reflects better the nature of the model; see the magnetic field, g is gravity acceleration, f is the Coriolis = ˆ below.) Its applications were further discussed in Dikpati parameter, f 2, and z is the angular velocity of rota- and Gilman (2001a) and Dikpati and Gilman (2001b). The tion. We also introduced the magnetic pressure: spectrum of linear waves and nonlinear stationary wave so- b2 lutions were established in Shecter et al. (2001), and Hamil- p∗ = p + ρ , (4) tonian structure and hyperbolicity properties were investi- 2 gated in Dellar (2002). The primary purpose of the present and considered, as usual, that the centrifugal effects are hid- paper is systematic derivation of the mRSW model from den in p. If the axis of rotation is supposed to be parallel to the full MHD equations by vertical averaging, which will the gravity acceleration, then f = f0 = const, which corre- (1) clarify the basic hypothesis underlying the model, and sponds to the f -plane approximation for the tangent plane (2) immediately give multi-layer generalizations. These gen- to the rotating planet/star for geo- and astrophysical applica- eralizations allow for incorporation of the baroclinic effects tions. For scales of the motion sufficiently small with respect in the model. Another purpose is to establish the quasi- to the radius of the planet/star, the non-verticality of the ro- geostrophic (QG) approximation of the (multi-layer) mRSW, tation axis may be taken into account in the β-plane approx- arising in the limit of strong rotation, as is traditionally imation: f = f0 + βy, where y is the latitudinal coordinate.

Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union. 894 V. Zeitlin: Remarks on rotating shallow-water magnetohydrodynamics

For what follows, it is important to note that the scaling for Again, the hypothesis that material surfaces are at the same the magnetic field is chosen in Eqs. (1)–(4) in such a way that time magnetic surfaces is crucial for arriving at such simple it is measured in velocity units. Effects of molecular dissipa- forms of integrated equations. It should be noted that the ver- tion and diffusion are neglected in Eqs. (1)–(3). If necessary, tical component of magnetic field b3 is not supposed to be they may be re-introduced via the standard viscosity, electro- zero, but in fact decouples from the mRSW equations to be conductivity and diffusion terms proportional to the Lapla- obtained below, and may be recovered once all other fields cians of velocity, magnetic field and density, respectively, in are determined, as is the case with v3 in the ordinary RSW the corresponding equations. equations. Equations (1)–(3) are written in the most general form. We Finally, we integrate Eq. (3) and get: should emphasize that the procedure below is applicable as z z z Z 2 Z 2 Z 2 well to their simplified versions, like the Boussinesq approx- ∂ dzρ + ∇ · dzρv = 0, ∇ · dzb = 0. (11) imation, where velocity is supposed to be divergenceless and t h h h h density is advected. Thermal effects may be introduced in z1 z1 z1 such models (cf. Rachid, 2008 in the solar tachocline con- 2.2 Mean-field approximation and magnetohydrostatic text) by relating variable parts of density to temperature. Fi- hypothesis nally, potential temperature θ may be used as a thermody- namical variable instead of density. Up to now, no approximation has been made whatsoever. The The horizontal momentum and magnetic field Eqs. (1) and only hypothesis was the existence of a pair of material sur- (2) may be rewritten with the help of Eq. (3) in the form of faces that are, at the same time, magnetic surfaces. Let us conservation laws: introduce the vertical averages R z2 dzF ∂t (ρv) + ∇ · (ρv ⊗ v)− F¯ = z1 , (12) − ∇ · (b ⊗ b) + gzˆ + f zˆ ∧ (ρv) + ∇p∗ = 0, (5) z2 z1 A · B ∂t (b) + ∇ · (v ⊗ b) − ∇ · (b ⊗ v) = 0. (6) and apply the mean-field approximation, i.e., replace by A¯ · B¯ for any A and B. We thus get from Eqs. (10)–(11): Here and below we use the shorthand notation ∇ ·
ρ¯ (z2 − z1) ∂t v¯h + v¯ · ∇hv¯h + f zˆ ∧ v¯h (A ⊗ B) ≡ ∂i (AiBk), i,k = 1,2,3 for a pair of vector fields  ¯ ¯  A,B. − ∇ (z2 − z1)bh ⊗ bh = We now proceed by vertical integration of the horizon- z Z 2 tal momentum Eq. (5) between a pair of material surfaces ∗ ∗ ∗ − ∇h dzp − ∇hz1 p + ∇hz2 p , (13) z1 z2 z1,2(x,y,t): z1 dzi  ¯   ¯  | = = + + = ∂t (z2 − z1)bh + ∇h · (z2 − z1)v¯h ⊗ bh v3 zi ∂t zi u∂xzi v∂yzi, i 1,2, (7) dt  ¯  − ∇h · (z2 − z1)bh ⊗ v¯h = 0, (14) which we suppose to be, at the same time, magnetic surfaces: ∂t [ρ¯ (z2 − z1)] + ∇h · [ρ¯ (z2 − z1)v¯h] = 0,  ¯ ∇h · (z2 − z1)b = 0. (15) b · ∇zi = 0. (8) Detailed discussion of the applicability of the mean-field With the help of the Leibnitz formula, we get: approximation is out of the scope of the present paper. It is R z2 R z2 obvious, however, that it requires sufficiently slow variations ∂t dzρvh + ∇h · dz(ρvh ⊗ vh) − z1 z1 of all fields in the vertical direction. As usual, corrections to ∇ · R z2 dz(b ⊗ b ) + f zˆ ∧ R z2 dz(ρv ) h z1 h h z1 h the mean-field theory may be accounted for by parameter- = −∇ R z2 ∗ − ∇ ∗| + ∇ ∗| izing neglected Reynolds stresses. Thus, following the tra- h z dzp hz1 p z hz2 p z , (9) 1 1 2 ditional line of argument, “turbulent” viscosity, diffusivity, where the index h denotes the horizontal part. This deriva- and conductivity may be introduced relating the neglected tion follows the standard procedure for obtaining the non- stresses to the mean fields. magnetic RSW equations, cf. Zeitlin (2007, Chapter 1). Note We will now make a crucial magnetohydrostatics hypoth- that additional surface terms appear if the hypothesis of mag- esis that will allow one to get the mRSW equations from the netic surfaces is relaxed. vertically averaged MHD in the mean-field approximation. We now integrate the horizontal magnetic field equations It consists of supposing vertical accelerations to be small, as in the same way and get: well as the term b·∇b3, and of replacing the vertical momen- z2 z2 z2 tum equation with the magnetohydrostatic balance relation: Z Z Z ∂t dzbh + ∇h · dz(vh ⊗ bh) − ∇h · dz(bh ⊗ vh) = 0. (10) ∗ z1 z1 z1 ρg = −∂zp . (16)

Nonlin. Processes Geophys., 20, 893–898, 2013 www.nonlin-processes-geophys.net/20/893/2013/ V. Zeitlin: Remarks on rotating shallow-water magnetohydrodynamics 895

Thus terms (with possibly non-constant “phenomenological” coef- ficients) if the hypotheses of turbulent viscosity and turbulent z Z ∗ ∗ ∗ magnetic diffusivity are applied to parameterize the stresses. p = −g dzρ + p ≈ −gρ¯ (z − z1) + p , (17) z1 z1 As already mentioned, thermal effects may be introduced z1 in the mRSW equations, following Ripa (1995) (cf. also Del- lar, 2003). If an additional equation of advection of temper- or ature (or potential temperature θ) is added to the original set z of 3-D equations, it will give the 2-D advection equation by Z 2 ∗ ∗ ∗ vertical averaging: p = +g dzρ + p ≈ gρ¯ (z2 − z) + p . (18) z2 z2 z ∂t θ + v · ∇θ = 0, (23) ¯ Traditionally, the mean density ρ is considered to be con- to be added to the system (19)–(22). Variable potential tem- stant in the RSW context, which will be our hypothesis perature in the buoyancy term in (1) will lead to the replace- below. Yet this hypothesis may be relaxed in the case of ∇ ∇ + 1 ∇ ment of the term g h by θ h 2 h θ after the vertical av- pure RSW, leading to the so-called Ripa’s equations Ripa eraging and proper renormalizations. (1995), i.e., shallow-water equations with variable mean den- One may extend the simplest mRSW system (19)–(22) by sity. mRSW equations including thermal effects may be ob- 1 superimposing N layers of different mean density, still un- tained along the same lines – see below. der the magnetohydrostatic hypothesis, ending up (at the top and at the bottom) either with a fixed (flat or not) or with a 2.3 Boundary conditions – multi-layer configurations free material surface. As a result, multi-layer mRSW models The final step consists of imposing boundary conditions at arise, allowing one to include the baroclinic phenomena in consideration. The structure of the multi-layer mRSW equa- z1,2. In the simplest configuration, one of the material sur- faces is fixed to be constant and the other is free with con- tions is clear from the above construction: they will inherit stant magnetic pressure above it, which leads to the mRSW the velocity and (magnetic) pressure terms from the multi- equations as proposed in Gilman (2000): layer RSW equations, with the same addition of a magnetic field in each layer as in Eq. (19). Mass is conserved layer- 1 wise. As to the magnetic field equations, they will be the ∂ v + v · ∇v + f zˆ ∧ v + g∇h = [∇ (hb ⊗ b))], (19) t h same as in Eqs. (21) and (22) for each layer. We give as an example the equations of the two-layer mRSW with fixed flat ∂t h + ∇ · (vh) = 0, (20) upper and lower boundaries at a distance H = const: ∇ · (hb) = 0 (21) 1 1 + · ∇ + ˆ ¯ + ∇ ∗ ∂t b + v · ∇b = [∇ (hv ⊗ b))], (22) ∂t vi vi vi f zρi πi h ρ¯i 1 where h = z − z , z = const, and we omit the bars and the = [∇ (hi bi ⊗ bi))] ,i = 1,2; (24) 2 1 1 h index h. Note that if for some reason the fixed material sur- i + ∇ · = + = face is not flat: z = η(x,y), “magnetic topography” η will ∂t hi (vihi) 0, h1 h2 H, (25) 1 ∗ ∗ enter the equations with a replacement h → h − η(x,y) ev- π1 = (ρ¯1 −ρ ¯2)gh1 + π2 , (26) erywhere except for the gravity term. Note also that in the ∇ · (hibi) = 0 (27) MHD context, unlike the standard RSW equations, the fixed 1 surface may be the upper one as well, with corresponding ∂t bi + vi · ∇bi = [∇ (hi vi ⊗ bi))], (28) h changes in the mRSW equations. i We should emphasize that the mRSW equations (19)–(22) where the subscripts i = 1,2 denote the lower and the up- are obtained under the hypothesis of no dissipation and strict per layer, respectively, no summation over repeated indices ∗ mean-field approximation. If molecular viscosity and mag- is supposed, πi are magnetic pressures in the respective lay- netic diffusivity are kept in the original equations (1)–(3), it ers, hi – thicknesses of the layers, the bottom topography is is easy to see that, through the above-described vertical av- not introduced, for simplicity, and the subscript h is omitted. eraging procedure, they would result in terms proportional The one-layer RSW model is recovered in the limit ρ¯2 → 0. to ∇2v and ∇2b (with two-dimensional ∇) in the r.h.s. of the Eqs. (19) and (22). As was already mentioned, the devia- tions from the strict mean-field theory would result in similar

1The relation between mRSW and Ripa’s equations, in partic- ular their common Hamiltonian structure, was discussed in Dellar (2003). www.nonlin-processes-geophys.net/20/893/2013/ Nonlin. Processes Geophys., 20, 893–898, 2013 896 V. Zeitlin: Remarks on rotating shallow-water magnetohydrodynamics

3 Properties of the barotropic mRSW system and the while the first ageostrophic correction acquires a magnetic QG limit addition:   (1) = + (0) · ∇ (0) +
3.1 General properties v2 ∂t v v1 J A,∂yA   (1) = − + (0) · ∇ (0) + The one-layer mRSW equations (19)–(22) can be rewritten v1 ∂t v v2 J (A,∂xA). (37) in conservative form: Here H is the mean depth of the layer, and h = H(1+Roη).   1 2 When plugged into (31), together with (33) this gives the ∂t (hv) + ∇ h (v ⊗ v − b ⊗ b) + gh 2 quasi-geostrophic MHD equations (QG MHD): + f hzˆ ∧ v = 0, (29) 1 ∂ ∇2η + J (η,∇2η) − ∂ η − J (A,∇2A) = 0, + ˆ ∧ ∇ ˆ · ∧
= t 2 t ∂t (hb) z hz (b v) 0, (30) Rd ∂t h + ∇ · (vh) = 0, (31) ∂t A + J (η,A) = 0. (38) ∇ · = (hb) 0, (32) In the limit of infinite deformation radius Rd → ∞, the equations in (38) become the standard 2d MHD. It is worth useful for numerical simulations. noting that, in non-magnetic RSW, the corresponding clas- Magnetic potential can be introduced for vertically inte- sical QG equation may be derived straightforwardly from grated horizontal magnetic fields, thus resolving the con- the PV equation, and expresses the Lagrangian conserva- straint (32): tion of PV. In Eulerian terms this is translated in terms of hb = zˆ ∧ ∇A,⇒ ∂t A + v · ∇A = 0. (33) conservation of the Casimir functionals – any function of 2 1 quasi-geostrophic PV defined as ∇ η − 2 η, if integrated Rd The magnetic field therefore may be eliminated in favor of over the domain of the flow, is conserved. Here, although A in Eqs. (19) and (29). The system thus becomes a RSW PV is not conserved, we still have QG MHD equations. In- system with additional forcing in the momentum equations, stead of a family of PV Casimirs, we now have two families due to the magnetic field that is determined from the pas- of Casimirs (Zeitlin, 1992): integrated functions of the mag- sively advected magnetic potential: netic potential, and integrated functions of magnetic potential 1  ∇A times quasi-geostrophic PV. ∂ v + v · ∇v + f zˆ ∧ v + g∇h = − zˆ ∧ J A, , (34) t h h If the β effect is introduced, with β of the order of Ro, as usual, the system (38) becomes to be completed with Eqs. (31) and (33). Here and below J 2 2 1 2 ∂t ∇ η + J (η,∇ η) − 2 ∂t η − J (A,∇ A) + β∂xη = 0, denotes the Jacobian. Rd As was repeatedly mentioned in the literature, cf. Dellar ∂t A + J (η,A) = 0. (39) (2002), the main difference between the RSW and mRSW systems is non-conservation of potential vorticity (PV) in the The system (39) with infinite deformation radius (and with latter. The only Lagrangian invariant, therefore, is the mag- addition of turbulent viscosity and conductivity) was recently netic potential A, cf. Eq. (33). introduced heuristically and was studied numerically in To- bias et al. (2007) in the context of solar tachocline. The same 3.2 Quasi-geostrophic approximation system, or rather its two-layer counterpart following from the system (24)–(28) and used below in Sect. 4, were recently The limit of strong rotation is characterized by the small- derived directly from the full MHD equations by Umurhan ness of both Rossby number Ro = U , and magnetic Rossby (2013). f0L B number Rom = , where U and B are the typical averaged f0L 3.3 Linear wave spectrum on the f - and β-planes, and velocity and magnetic field in the fluid layer, and L is a typ- the role of external magnetic fields ical horizontal scale. Supposing that both Rossby numbers are of the same order of magnitude, and applying the stan- The QG version of the mRSW equations was derived above dard straightforward expansion in a semi-heuristic manner, by using the slow time scale, v = v(0)(x,y,T ) + Rov(1)(x,y,T ) + O(Ro2) (35) and thus filtering “by hand” the fast inertia–gravity waves. √ A more formal justification of this limit via the separation for motions of the scale L ∝ R = gH depending only on of fast and slow variables and systematic fast-time averaging d f0 slow time T ∝ Rot, cf. Zeitlin (2007), we get that the leading may be achieved, following the lines of Reznik et al. (1992) order velocity field is, as it should be, geostrophic: and Zeitlin et al. (1992). The situation, however, will be more tricky in the presence of the external magnetic field. Indeed, (0) = (0) = − v2 ∂xη, v1 ∂yη, (36) it it easy to see that, due to its Lagrangian conservation char- acter, the magnetic field equation (33) is “slow” and does not

Nonlin. Processes Geophys., 20, 893–898, 2013 www.nonlin-processes-geophys.net/20/893/2013/ V. Zeitlin: Remarks on rotating shallow-water magnetohydrodynamics 897 change the linear, i.e., inertia–gravity, wave spectrum of the RSW system in the absence of background magnetic fields. d(0) h i Thus, the procedure of Reznik et al. (1992) and Zeitlin et al. i ∇2π ∗ − (−1)iD−1η − J (A ,∇2A ) = 0, t i i i i (1992) should work for perturbation about the rest state with- d (0) out background magnetic fields. However, the Alfvén waves di Ai = 0, i = 1,2, (42) that should appear if a background magnetic field is present, dt do not have a spectral gap, see below, and may invalidate the where slow-fast separation in this case. This issue will be addressed (0) di ∗
in detail elsewhere. (...) := ∂t (...) + J πi ,... , i = 1,2. (43) We recall that the linearization of Eqs. (19)–(22) about dt the state of rest h = H on the f -plane f = const, with con- Here Di are nondimensional thicknesses of the layers, and stant magnetic field b = B, results in harmonic wave solu- η denotes a nondimensional interface deviation. In the stan- tions with wavefrequency ω and wavevector k satisfying the dard in GFD limit ρ2−ρ1 → 0 it is simply expressed in terms following dispersion relation, cf. Shecter et al. (2001): of the pressure difference: η = π2 − π1. No summation over repeated indices is understood, and stars at πi are omitted. 2 2 gHk + f 1q 2 The details of the scaling and the procedure may be found in ω2 = (B · k)2 + ± gHk2 + f 2
+ 4f 2(B · k)2, (40) 2 2 Zeitlin (2007, Chapters 1 and 2). Their extension to the MHD case is straightforward. which gives, in the limit of no rotation, ω2 = (B·k)2, ω2 = On the β-plane, the terms β∂ π should be added in (B · k)2 + gHk2, i.e., Alfvén and mixed Alfvén–gravity x i Eq. (42), and formal linearization of Eq. (42) will give baro- waves. clinic (and barotropic) mixed Alfvén–Rossby waves in the Likewise the formal linearization of the system (39) over presence of a background magnetic field. It is well known, the rest state with constant background magnetic field B : however, that in the baroclinic system Rossby waves can η = 0,A = B y − B x results in harmonic solutions with 1 2 also propagate due to the velocity shear between the lay- wavefrequency ω and wavevector k = (k ,k ) with the fol- 1 2 ers even in the absence of β. For QG motions described by lowing dispersion relation: Eq. (42), a shear corresponds to an interface inclination via v the geostrophic balance. It is also well known that on the u !2 βk u βk f -plane, any shear is unstable for sufficiently long wave per- ω = − 1 ± t 1 + (B · k)2, (41) 2 + −2 2 + −2 turbations due to the baroclinic instability (on the β-plane a 2(k Rd ) 2(k Rd ) threshold for the instability exists), cf. Pedlosky (1982). Let corresponding to mixed Alfvén–Rossby waves. Indeed, in us see how incorporation of the magnetic field influences the the limit of vanishing magnetic field (41) gives the usual baroclinic instability. By linearizing about the state Rossby waves, and in the opposite limit of vanishing β it ∗ π = −Uiy, Ai = Biy, (44) gives the Alfvén waves. i It is clear from Eq. (41) that, in the presence of the with constant Ui, Bi, and looking for harmonic solutions background magnetic field, the frequency spectrum is not with wavefrequency ω and wavenumber k in the strongly de- bounded from above anymore, and the formal validity of the generate, but simplest to analyze case D1 = D2 = D, U1 = system (39) as “slow” limiting equations of the full mRSW −U2 = U, we arrive at the following dispersion relation: equations remains to be proved. s (1 + B2 + B2)k2 + D−1(B2 + B2 − 2) In any case, the known effect of the “elasticity” of the mag- = ± 1 2 1 2 ω Uk1 − . (45) netic field leading to Alfvén waves is well represented in the k2 + 2D 1 barotropic mRSW and its “slow” version. We will see that The standard baroclinic instability result Pedlosky (1982) is this effect will play a stabilizing role when baroclinic effects recovered in the limit B1,2 → 0. are included. Thus, if the magnetic field is strong enough in any layer, the baroclinic instability disappears. Such a stabilizing effect of the magnetic field could be anticipated due to its “elastic- 4 Effects of baroclinicity ity” mentioned above. It is characteristic, in fact, not only of The multi-layer mRSW can be treated in a similar way as the large-scale slow-evolving geostrophic baroclinic instabil- the one-layer case, by introducing magnetic potentials for ity, but also of the rapid ageostrophic instabilities such as the each layer. The QG approximation may be as well developed, Kelvin–Helmholtz one, which can be easily studied in the again along the standard lines, giving the following 2-layer framework of the non-rotating 2-layer equations (24)–(28) QG MHD equations for (magnetic) pressures and magnetic by linearizing about the basic state with horizontal velocity potentials in the layers in the f -plane approximation: shear between the layers. Although the magnetic field does not cure the instability, its influence is stabilizing, adding up with gravity (not presented). www.nonlin-processes-geophys.net/20/893/2013/ Nonlin. Processes Geophys., 20, 893–898, 2013 898 V. Zeitlin: Remarks on rotating shallow-water magnetohydrodynamics

5 Concluding remarks

Thus, we have shown how the mRSW model of Gilman The publication of this article (2000) arises from vertical averaging of the MHD equa- is financed by CNRS-INSU. tions for the rotating fluid in the gravity field, and applica- tion of the mean-field and magnetohydrostatic hypotheses. Multi-layer generalizations easily follow from this construc- tion. Although it is important to keep in mind the limits of References applicability of the mRSW related to these hypotheses, the example of non-magnetic RSW and its applications in geo- Bouchut, F.: Ch. 4, in: Nonlinear Dynamics of Rotating Shallow physical fluid dynamics show that such models remain use- Water: Methods and Advances, edited by: Zeitlin, V., Elsevier, ful far beyond their formal validity range. The added value, NY, 2007. with respect to the pioneering work of Gilman (2000), is that Dellar, P. J.: Hamiltonian and symmetric hyperbolic structure of we show that the mRSW equations arise universally from shallow water magnetohydrodynamics, Phys. Plasmas, 9, 1130– the vertical averaging of basic MHD and independently of 1136, 2002. Dellar, P.: Common Hamiltonian structure of the shallow water the details of stratification and compressibility. The method equations with horizontal temperature gradients and magnetic also gives the possibility to include thermal effects and shows fields, Phys. Fluids, 15, 292–297, 2003. how the deviations from the strict mean-field approximation Dikpati, M. and Gilman, P. A.: Prolateness of the solar tachocline may be accounted for with turbulent viscosity and diffusivity inferred from latitudinal force balance in a magnetohydrody- hypotheses. namic shallow water model, Astrophys. J., 552, 348–353, 2001a. The “balanced” quasi-geostrophic limit of the mRSW with Dikpati, M. and Gilman, P. A.: Flux-transport dynamos with alpha- filtered inertia–gravity waves was established in a semi- effect and global instability of tachocline differential rotation: a heuristic manner and is closely related to 2d MHD. On the solution for magnetic parity selection in the sun, Astrophys. J., β-plane it gives a framework for studying mixed Alfvén– 559, 428–442, 2001b. Rossby waves, although the formal validity of the QG MHD Gilman, P. A.: Magnetohydrodynamic “shallow water” equations as the slow limit of the mRSW in the presence of constant for the solar tachocline, , Astrophys. J. Lett, 544, L79–L82, 2000. Pedlosky, J.: Geophysical Fluid Dynamics, Springer, NY, 1982. magnetic field remains to be proved. As follows from the re- Rachid, F. Q., Jones, C. A., and Tobias, S. M.: Hydrodynamic in- sults of Sect. 3.3, sufficient smallness of the magnetic field stabilities in the solar tachocline, Astronomy Astrophysics, 488, may be necessary for such a proof. 819–827, 2008. The parallels between RSW and mRSW may be devel- Reznik, G., Zeitlin, V., and BenJelloul, M.: Nonlinear theory of oped further. For instance, the existence of mixed Alfvén– geostrophic adjustment. Part 1. Rotating shallow water model, equatorial waves may be straightforwardly established for J. Fluid Mech., 445, 93–120, 2001. the mRSW equations on the equatorial β-plane where f0 = Ripa, P.: On improving a one-layer ocean model with thermody- 0. These waves may be important in astrophysical applica- namics, J. Fluid Mech., 303, 169–207, 1995. tions. Mixed Alfvén–Kelvin waves can be found propagating Shecter, D. A., Boyd, J. F., and Gilman, P. A.: Shallow-water mag- along the boundary parallel to the magnetic field, a configu- netohydrodynamic waves in the solar tachocline, Astrophys. J. ration that may exist in the laboratory, and so on. Lett, 551, L185–L188, 2001. Tobias, S. M., Diamond, P. H., and Hughes, D. W.: Beta-plane mag- Let us finally mention that an important advantage of netohydrodynamic turbulence in the solar tachocline, Astrophys. mRSW equations is that, in their conservative form (29)– J. Lett., 667, L113–L116, 2007. (32) (an analogous form may be easily established for a Umurhan, O. M.: The equations of magnetogeostrophy, multi-layer mRSW with free upper boundary), they may arXiv:1301.0285v1, 2013. be efficiently treated numerically with the help of modern Zeitlin, V.: On the structure of phase-space, Hamiltonian vari- finite-volume methods, cf. Bouchet (2007). This work is in ables and statistical approach to the description of 2-dimensional progress. hydrodynamics and magnetohydrodynamics, J. Phys A, L171– L175, 1992. Zeitlin, V. (Ed.): Nonlinear Dynamics of Rotating Shallow Water: Acknowledgements. This work was supported by the French ANR Methods and Advances, Elsevier, NY, 2007. grant “SVEMO”. Zeitlin, V., Reznik, G., and BenJelloul, M.: Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously strat- Edited by: V. Shrira ified primitive equations, J. Fluid Mech., 491, 207–228, 2003. Reviewed by: O. Umurhan and one anonymous referee

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