2.1 Non-dimensionalisation

To identify the parameters involved, it is helpful to write the governing equations in a non-dimensional form. The external input to the system is the vorticity source term $s(x,y,t)$ , and we take it to have magnitude $\mathscr{S}$ and act on a spatial scale $\mathscr{L}$ . We choose to use the length scale $\mathscr{L}$ and the time scale $\mathscr{T}\equiv \mathscr{S}^{-1/2}$ to non-dimensionalise the problem, rescaling quantities such as

This recovers (2.1)–(2.4) in a dimensionless form, adorned with daggers. The system is then characterised by the dimensionless parameters, the Grashof number $\mathit{Gr}$ and the source Rhines number $\mathit{Rh}_{s}$ , given by

as well as Prandtl and magnetic Prandtl numbers giving the ratios of the appropriate diffusivities, and the functional form of the source term $s$ . It should be noted that $\mathit{Rh}_{s}$ is here based on the source strength rather than on the actual flow velocities realised, whereas the usual Rhines number $\mathit{Rh}=\mathscr{U}/{\it\beta}\mathscr{L}^{2}$ , like the Reynolds number $\mathit{Re}=\mathscr{L}\mathscr{U}/{\it\nu}$ , would be a diagnostic depending on the flow velocities $\mathscr{U}$ that emerge in the forced system (Rhines Reference Rhines1975).

As a key simplification in this study we will take all the diffusivities to be equal,

so the Prandtl numbers are unity, and we use the neutral quantity ${\it\lambda}^{\dagger }$ to denote any diffusivity or viscosity in what follows. This is both to make analytical progress and to reduce the number of parameters involved, making our study more manageable. For ease of notation it is also convenient to use the dimensionless parameters $\{{\it\lambda}^{\dagger },{\it\beta}^{\dagger }\}$ in what follows rather than $\{\mathit{Gr},\mathit{Rh}_{s}\}$ . From now on we will also drop the daggers on non-dimensional quantities, and so the key parameters at the outset are simply $\{{\it\lambda},{\it\beta}\}$ , with the source $s$ taken of length scale and magnitude of order unity.

2.2 Quasi-linear approximation

Table 1.

Mean and fluctuating fields (the second of each pair has had the pure advection term $\text{e}^{-\text{i}mU(y)t}$ removed).

We adopt the quasi-linear approximation in which we take a mean or background flow and magnetic field independent of $x$ , namely

and assume that these are quasi-steady, varying on a longer time scale than that of the fluctuating fields, namely those fields with a non-trivial $x$ -dependence. We need then to define the source term $s$ that creates disturbances, evolving according to linear dynamics, linearised about the background profile (2.8). We take first the specific case of a delta-function (in time) source of vorticity that is a single Fourier mode in  $x$ ,

In the quasi-linear approximation we break fields into mean and fluctuating components, as summarised in table 1,

and retain only the leading harmonics shown. We refer to an evolving $\text{e}^{\text{i}mx}$ disturbance as a wave (even in regimes where it may be heavily damped), and this obeys linear equations

with initial conditions from (2.9),

A prime here denotes the $y$ -derivative of any mean field, these being related as shown in table 1.

The feedback on the mean fields in (2.10) from the waves via the quadratic terms in (2.1)–(2.3) is retained, and this can be written in terms of fluxes

from Reynolds and Lorentz stresses in (2.16), and from advection of vector potential and scalar fluctuations in (2.17). Over the lifetime of a wave, the feedback on the mean flow from any one of these fluxes is given by the integral

Here and elsewhere we find it convenient to use $\mathit{Z}$ as a place-holder: it can be $R$ , $L$ , $M$ or $P$ as in (2.16), (2.17), or involve further symbols later on.

We have (2.11)–(2.15) for a single wave launched at $t=0$ and the feedback (2.16)–(2.18) on the mean fields as the wave evolves and dissipates. As this effect must be weak in any quasi-linear set-up, we now incorporate a stream of such waves, replacing (2.9) by the renewing source

in which at each unit of time vorticity is introduced into the flow with random independent phases ${\it\mu}_{j}$ uniformly distributed over $[-{\rm\pi},{\rm\pi}]$ . Given the random phases, in a time average (or ensemble average) all waves contribute independently to the quadratic fluxes. The equations for the mean fields then become

where it is convenient to combine Reynolds and Lorentz terms via

It should be noted that the term $\mathscr{F}_{\mathit{K}}$ would also appear in a mean momentum equation, as per (3.18) below, but we usually prefer to remain within a vorticity–stream-function formulation.

2.3 Summary and comments

The quasi-linear system that we will consider is well established in the literature and consists of solving (2.11)–(2.15) for general mean profiles $U$ , $B$ and ${\it\Sigma}$ , and then determining the feedback on each mean field through the fluxes (2.16)–(2.18) that feed into (2.20)–(2.22). Several comments are in order.

(i) Given the renewing source (2.19), (2.20)–(2.21) for the mean fields are only valid on time scales $t\gg 1$ , that is over many waves launched into the flow. Furthermore, the quasi-linear approximation (as we use it) keeps the mean fields $U$ and $B$ steady while we solve (2.11)–(2.15) and compute the integrated fluxes in (2.18). There is thus a key consistency condition, that the time scale of evolution of the mean fields in (2.20)–(2.22) be greater than that over which the waves contribute to the integrated fluxes (2.18), as stressed by Bouchet & Morita (Reference Bouchet and Morita2010). This consistency condition needs to be assessed as any quasi-linear model is developed.

(ii) One of the features of this study is keeping the source term (2.19) as tractable as possible without too much loss of generality. Regarding this, we note that we could have introduced a more general time dependence, for example involving a stationary random process with a given correlation function (decaying on a time scale shorter than that of the evolution of the mean fields), for example delta-correlated. However, doing so would increase complexity for us without adding greatly to generality. In fact, we can note that the problem as formulated in (2.11)–(2.15) gives the Green’s function for waves, and the effect of a more general time dependence of the source $s$ could thus be obtained by integration.

(iii) Likewise, we have considered a single source mode with wavenumber $m$ in (2.19); we treat $m$ as a parameter (though strictly it could be scaled to $m=1$ in our non-dimensionalisation). More generally, we could consider modes of the form $\text{e}^{\text{i}mx+\text{i}ny}$ and arbitrary sums of such modes, for example forming an isotropic or ring forcing as used in many other studies. However, because of shearing in the gradients $U^{\prime }$ of the mean flow, a more general mode $\text{e}^{\text{i}mx+\text{i}ny}$ simply corresponds to a shift in time of the mode $\text{e}^{\text{i}mx}$ , and so our analysis could be extended, creating additional complexity in integrating over angles but without any fundamental changes. We should remark though that our forcing is anisotropic, in contrast to some other studies, and this is known to enhance instabilities leading to zonal flows, by selecting a preferred direction in the system, as discussed in Srinivasan & Young (Reference Srinivasan and Young2012) and Bakas & Iouannou (Reference Bakas and Iouannou2013).

(iv) We have not discussed the strength of the background flow $U$ , scalar field ${\it\Sigma}$ or magnetic field $B$ . In fact, the status of these is rather different. A background flow $U$ can grow from a seed through zonostrophic instability, and so we cannot set its scale at the outset. For passive scalar transport, the magnitude of the background gradient ${\it\Sigma}^{\prime }$ is a constant that can be set arbitrarily. The strength of a mean magnetic field (that is, $B$ averaged over the $y$ -coordinate) is another constant that must be set initially and will introduce a further parameter into the problem. It should be noted that given our present non-dimensionalisation (2.5), the magnetic field strength $B$ is measured relative to the velocity strength, corresponding to an inverse magnetic Mach number.

(v) Our primary focus will be on ${\rm\nabla}^{2}$ dissipation (we refer to this as molecular dissipation) in the development from (2.1)–(2.3), but we will below remark briefly on more general forms of dissipation, using $-{\it\lambda}_{r}(-{\rm\nabla}^{2})^{r}{\it\omega}$ in (2.1), and likewise for $a$ and ${\it\sigma}$ . The case $r=0$ is bottom drag, $r\geqslant 2$ is hyperviscosity (or hyperdiffusion) and $r=1$ , ${\it\lambda}_{1}\equiv {\it\lambda}$ is standard viscous dissipation. We stress that these are the ‘molecular’ effects – in other words terms that we introduce into the governing equations, to be distinguished from the ‘effective’ transport effects, for example ${\it\nu}_{\mathit{eff}}$ , ${\it\kappa}_{\mathit{eff}}$ , that emerge from our analysis below.

3.1 Multiple-scale formulation

We first apply a transformation to absorb the advective term,

and obtain

with $\mathscr{L}$ denoting the Laplacian operator,

The flux (2.16) may then be written as

It should be noted that the leading-order term in ${\it\zeta}$ from (3.4), which is a purely real multiple $m^{2}(1+{\it\Omega}^{2}t^{2})$ of ${\it\phi}$ , does not contribute to the flux $F_{\mathit{R}}$ . This absence of a phase shift between the two fields means that a vorticity disturbance on an exactly linear shear flow ${\it\Omega}=\text{const.}$ cannot contribute a mean Reynolds stress: it is higher-order terms that need to be included, involving gradients of  ${\it\Omega}$ .

The development so far is exact (within the quasi-linear approximation), and to make further progress we take the case when the mean or background flow varies on a large scale $Y$ , setting

with ${\it\varepsilon}\ll 1$ . We now write ${\it\Omega}=-U^{\prime }$ , using the prime to denote a $Y$ -derivative of any background field. Here and elsewhere we suppress the (slow) time dependence of the background field, governed by (3.12) below. (It should be noted that the correlation function approach of Srinivasan & Young (Reference Srinivasan and Young2012) does not assume scale separation.)

We now adopt the scalings

These give for the wave evolution on the $T$ time scale

with

The equation (2.20) for mean fields takes the form

and these fields develop on the long time scale $\mathscr{T}={\it\varepsilon}^{2}T$ , where as a consequence of the new scalings in (3.8), the equation

now replaces (2.18) for any label  $\mathit{Z}$ .

Equations (3.9)–(3.13) are the ones we wish to expand and study asymptotically for small  ${\it\varepsilon}$ . The rationale for the above choice of scaling in (3.7), (3.8) is to retain as many terms at leading order as possible in (3.9)–(3.13). It should be noted that from (3.7), the strength of the waves, and so the strength of the original driving force, is being reduced as ${\it\varepsilon}\rightarrow 0$ , to balance the viscous term in (3.12) for the mean fields. For more general dissipation the scalings would be as above except for

and give rise to the same set of equations with only modifications to the definition of $\mathscr{L}$ in (3.10). For $r=0$ (linear drag) there is no separation of scales between $\mathscr{T}$ and $T$ , and so the consistency condition discussed in the first comment of § 2.3 would not be satisfied: mean flow and waves would be damped on the same time scale (at least, unless some further limit were taken, or perhaps some combination of damping effects employed).

3.2 Numerical solution

Figure 1.

Simulation of waves with ${\it\beta}=0$ , $m=1$ , ${\it\varepsilon}=0.1$ , ${\it\lambda}=0$ and $U=\cos Y$ . In (a)  $\text{Re}\,{\it\omega}$ , (b)  $\text{Re}\,{\it\psi}$ and (c)  $F_{R}$ are plotted in the $(T,Y)$ -plane. In (d)  $F_{R}(Y,T)$ is shown as a function of $Y$ for $T=0.1,0.2,\ldots ,1$ (solid, innermost to outermost curve), with $U(Y)$ (dashed).

Before undertaking further analysis, it is worth gaining some intuition by simulating the system we have so far for the linear waves ${\it\omega}(Y,T)$ , ${\it\psi}(Y,T)$ . The system may be written as

equivalent to (3.9), (3.10), with here

A run with ${\it\beta}=0$ , ${\it\varepsilon}=0.1$ and $U=\cos Y$ is shown in figure 1: (a,b) show the real part of the vorticity ${\it\omega}$ and stream function ${\it\psi}$ respectively, in the $(T,Y)$ -plane. We have taken zero viscosity, ${\it\lambda}=0$ ; otherwise we would simply see a damping effect. Away from the maxima of $|U|$ at $Y=0$ , $\pm {\rm\pi}$ (where ${\it\Omega}=0$ ) the vorticity gains finer scales and the stream function is suppressed. The flux $F_{\mathit{R}}$ is shown in (c) and develops (red) peaks and (blue) troughs for large  $T$ .

The short-time evolution of flux $F_{\mathit{R}}(Y,T)$ is shown in (d) for $T=0.1,0.2,\ldots ,1$ (solid), going from the innermost (smoothest) curve to the outermost one. Figure 1(d) contains key information as we can write from (3.12) for the mean velocity

and here $U=\cos Y$ is shown dashed in figure 1(d). Thus, the short-time effect of the flux $F_{\mathit{R}}$ (as it contributes to $\mathscr{F}_{\mathit{R}}$ ) is to reinforce the original flow $U$ , which is precisely the effect leading to zonostrophic instability. If there is sufficient viscous damping, then only the short-time feedback is relevant, and below we will recover this effect, which naturally is well documented in the literature (e.g. Srinivasan & Young Reference Srinivasan and Young2012).

For greater times, the feedback from the flux $F_{\mathit{R}}$ becomes more oscillatory in $Y$ , changing sign in places with respect to the original $U=\cos Y$ profile, as seen in figure 1(c,d). This is investigated further below, but there is an indication that the effect of the feedback is to sharpen any pre-existing profile, increasing the flow where ${\it\Omega}=-U^{\prime }=0$ and flattening the flow profile where $U=0$ and $|{\it\Omega}|$ is maximal. It should be noted that a tendency to sharpen shear profiles is found in weakly damped vortex dynamics simulations by Dritschel & McIntyre (Reference Dritschel and McIntyre2008), and by Kim & MacGregor (Reference Kim and MacGregor2003) in a study of gravity waves in stratified fluid, together with a bifurcation to an oscillatory state.

3.3 Multiple-scale expansion

We now return to the development in § 3.1 and approximate. We expand quantities in powers of ${\it\varepsilon}$ , for example,

and the leading-order equations (3.9), (3.10) take the form

The accelerated decay through the $T^{3}$ term in the exponential is known as the shear–diffuse mechanism (Bernoff & Lingevitch Reference Bernoff and Lingevitch1994) responsible for flux expulsion in kinematic magnetic field evolution (Weiss Reference Weiss1966). This gives a first-order ordinary differential equation (ODE) with respect to $T$ , with the solution

The expansion gives us functions of ${\it\Omega}T$ rather than $T$ , and so it is convenient to call this ${\it\tau}$ and to try to reduce the number of parameters involved. We set

in terms of which we have the more compact form

We will for simplicity and without loss of generality consider points $Y$ in the profile where ${\it\Omega}>0$ , so that ${\it\tau}\in [0,\infty ]$ ; the final results are nonetheless correct also for ${\it\Omega}<0$ . Behaviour at points where ${\it\Omega}=0$ can be established by letting ${\it\Omega}\rightarrow 0$ ; although this need not detain us here, we make some further comments and qualifications in appendix A.

We then obtain the leading-order flux from (3.11) as a function of time,

(we have simply dropped the term of order ${\it\varepsilon}$ in (3.11), but not further complicated the notation), with

The leading-order integral may be written as

where we set, for any label $\mathit{Z}$ ,

It should be noted that $h_{\mathit{R}}=h_{\mathit{R}}(\hat{{\it\lambda}})$ only, and in fact $\hat{{\it\beta}}$ has dropped out of the calculations. The large-scale equation is now

with an effective viscosity ${\it\nu}_{\mathit{eff}}({\it\Omega},{\it\lambda})$ identified and arising from the externally forced waves. This term emerges from the two-scale analysis with the expected form, in other words giving transport of the large-scale vorticity field ${\it\Omega}(Y,\mathscr{T})$ . It should be noted, however, that although the underlying molecular viscosity ${\it\lambda}$ is always positive and so dissipative in effect, given that we have averaged over the vorticity equation with an imposed body force present, there is no reason why the effective (or ‘renormalised’) viscosity ${\it\nu}_{\mathit{eff}}$ need be positive, and generally it is not, this being the origin of zonostrophic instability in this system.

To summarise, (3.24) gives the instantaneous feedback $F_{\mathit{R}}$ from the wave to the mean flow, coded in the function $f_{\mathit{R}}({\it\tau},\hat{{\it\lambda}})$ in (3.25) with the damping factor $D^{2}$ . The integrated effect is found in $h_{\mathit{R}}(\hat{{\it\lambda}})$ and gives the effective viscosity in (3.26); the parameter $\hat{{\it\beta}}$ plays no role at this order (cf. Srinivasan & Young Reference Srinivasan and Young2012, Reference Srinivasan and Young2014). Figure 2(a) shows $D^{2}f_{\mathit{R}}$ as a function of ${\it\tau}$ in the inviscid limit, $\hat{{\it\lambda}}=0$ , and for increasing values of $\hat{{\it\lambda}}$ . Focusing on the inviscid case (outermost curve), it is worth noting that the feedback changes sign at ${\it\tau}=3^{-1/2}$ . The effect of increasing the damping, $\hat{{\it\lambda}}>0$ (inner curves), is primarily to suppress $f_{\mathit{R}}$ at ever earlier times. Figure 2(b) shows $h_{\mathit{R}}(\hat{{\it\lambda}})$ , which is the total integrated flux. We have $h_{\mathit{R}}(0)=1$ while for small $\hat{{\it\lambda}}$ the effective viscosity is positive, corresponding to the damping of the mean flow. At $\hat{{\it\lambda}}\simeq 0.95$ , $h_{\mathit{R}}$ changes sign, and for larger $\hat{{\it\lambda}}$ it is negative and so destabilising. For large $\hat{{\it\lambda}}$ the integral (3.27) is effectively cut off at times ${\it\tau}=O(\hat{{\it\lambda}}^{-1})$ and we can expand the functions $D$ and $f_{\mathit{R}}$ as power series in ${\it\tau}$ , giving

also shown in figure 2(b) (dashed curve).

Let us now relate this detailed calculation back to the bigger picture, for example as in the simulations in figure 1. For good scale separation, ${\it\varepsilon}\ll 1$ , at each $Y$ the flow is locally shearing, as given by the value of ${\it\Omega}(Y)$ . The subsequent evolution of a wave is determined by the key parameter $\hat{{\it\lambda}}$ and occurs on a time scale ${\it\tau}$ ; see (3.22). The evolution on the ${\it\tau}$ time scale is damped by the factor $D$ , which occurs at ${\it\tau}=O(\hat{{\it\lambda}}^{-1})$ for large $\hat{{\it\lambda}}$ , or at ${\it\tau}=O(\hat{{\it\lambda}}^{-1/3})$ for small $\hat{{\it\lambda}}$ , by the shear–diffuse mechanism. For small ${\it\varepsilon}$ , this is all that is needed to recreate the evolution of $F_{\mathit{R}}$ in figure 1(c,d), and for each $Y$ good agreement can be shown between the evolution in time $T$ in this figure, and in the analysis above based on the local ODE (3.20).

Figure 2.

(a) Plot of $D^{2}f_{\mathit{R}}$ as a function of ${\it\tau}$ for $\hat{{\it\lambda}}=0,0.1,0.2,0.4,\ldots ,3.2$ (from outermost to innermost curve). (b) Plot of $h_{\mathit{R}}$ as a function of $\hat{{\it\lambda}}$ (solid) and the approximation (3.29) (dashed).

It should be noted that in a profile with varying ${\it\Omega}$ , the local parameter $\hat{{\it\lambda}}$ will also vary. Taking ${\it\Omega}$ bounded, if ${\it\lambda}\gg 1$ then the waves are always in a viscous regime, $\hat{{\it\lambda}}\gg 1$ , whereas if ${\it\lambda}$ is small, at different locations $Y$ the waves may effectively be in a low-viscosity, $\hat{{\it\lambda}}\ll 1$ , or a viscous, $\hat{{\it\lambda}}\gg 1$ , regime. In a viscous regime we have

This destabilising negative viscosity term corresponds to the instability of the original steady Kolmogorov flow discussed in Meshalkin & Sinai (Reference Meshalkin and Sinai1961), and follows from the anisotropy of the forcing as discussed, for example, in Srinivasan & Young (Reference Srinivasan and Young2014). The inverse power of ${\it\lambda}$ shows the effect of large viscosity in cutting off the feedback on the mean flow, corresponding to capturing the short-time behaviour seen in figure 1(d). This result is key to zonostrophic instability, in which ${\it\Omega}\simeq 0$ initially everywhere. No matter what the molecular value of ${\it\lambda}>0$ is, the limit $\hat{{\it\lambda}}\gg 1$ and the result (3.30) are then appropriate, with the destabilising effect of a negative effective viscosity arising from the negative values of $f_{\mathit{R}}$ for short times and so of $h_{\mathit{R}}$ for large $\hat{{\it\lambda}}$ ; see figure 2.

We remark that even if we applied molecular hyperviscosity (i.e. hyperdiffusion) to the waves (end of § 2.3), which would lead to modified forms of $D$ and $\mathscr{L}$ , we would still recover effective viscosity for the large-scale fields, in other words via a term of the form $\partial _{Y}({\it\nu}_{\mathit{eff}}\partial _{Y}{\it\Omega})$ . Whereas negative effective viscosity leads to an ill-posed problem if it is only balanced by weaker positive molecular viscosity, having molecular hyperviscosity recovers a sensible problem for the large-scale fields, for example to integrate numerically. We leave this for future study, noting that molecular hyperviscosity should only be introduced with considerable caution: see Sukoriansky et al. (Reference Sukoriansky, Galperin and Chekhlov1999) in the case of turbulence and Zhang & Jones (Reference Zhang and Jones1997) for rapidly rotating convection.

3.4 Zonostrophic instability

At the end of the last section we indicated the origin of zonostrophic instability through the negative sign in (3.30). We study this in the more general context of a uniform background shear flow. Consider again (3.28) and think of ${\it\nu}_{\mathit{eff}}={\it\nu}_{\mathit{eff}}({\it\Omega},{\it\lambda})$ in general. A solution is simply steady uniform vorticity ${\it\Omega}(Y,\mathscr{T})={\it\Omega}_{0}=\text{const.}$ , that is uniform shear $U=-{\it\Omega}_{0}Y$ , and we can investigate the stability of this solution to perturbations.

We replace ${\it\Omega}\rightarrow {\it\Omega}_{0}+{\it\Omega}$ in (3.28) with now ${\it\Omega}\ll 1$ to give

at leading order. Seeking a normal mode in ${\it\Omega}$ proportional to $\exp (PK^{2}\mathscr{T}+\text{i}KY)$ gives the (scaled) growth rate $P$ as

The last term $-{\it\lambda}$ is the damping effect of molecular viscosity, and the nature of any instability is again clear as the result of a negative effective viscosity, taking place if ${\it\nu}_{\mathit{eff}}$ is negative and large enough. We first consider when ${\it\Omega}_{0}=0$ , in which case we insert the large- $\hat{{\it\lambda}}$ expansion (3.30) to obtain

We have zonostrophic instability if $P>0$ , which amounts to the condition $m^{2}{\it\lambda}<2^{-1/3}$ .

Figure 3.

Contour plot of $m^{2}P$ (3.32) in the $(m^{2}{\it\lambda},{\it\Omega}_{0})$ -plane. Contours increase from $m^{2}P=0$ (outermost) in steps of 0.1 to $m^{2}P=10$ (bottom left corner).

For the more general case of a uniform background vorticity distribution ${\it\Omega}_{0}\neq 0$ , the relative size of ${\it\Omega}_{0}$ and ${\it\lambda}$ is important through the value $\hat{{\it\lambda}}=m^{2}{\it\lambda}/{\it\Omega}_{0}$ in (3.32). We cannot evaluate $h_{\mathit{R}}$ analytically in general, but we can easily do so numerically, and figure 3 shows a contour plot of $m^{2}P$ in the $(m^{2}{\it\lambda},{\it\Omega}_{0})$ -plane. The outer contour gives $P=0$ , and outside this $P<0$ . Along the horizontal axis ${\it\Omega}_{0}=0$ the result (3.33) holds. For increasing ${\it\Omega}_{0}$ , zonostrophic instability is suppressed (Srinivasan & Young Reference Srinivasan and Young2014; Hsu & Diamond Reference Hsu and Diamond2015), being restricted to a finite range of viscosities $m^{2}{\it\lambda}$ bounded above zero, and turns off completely for ${\it\Omega}_{0}$ greater than approximately 0.25. It should be noted that the suppression of instability by shear is also evident in the sign of the second correction term in (3.29). The value of $m^{2}P$ diverges as ${\it\lambda}\rightarrow 0$ , ${\it\Omega}_{0}\rightarrow 0$ (bottom left, where the contours accumulate), and in fact the contours accumulate near the origin on the line given by $m^{2}{\it\lambda}/{\it\Omega}_{0}\simeq 0.95$ , on which $h_{\mathit{R}}$ changes sign.

3.5 Equilibrium profiles

We cannot readily time step (3.28) numerically: as ${\it\nu}_{\mathit{eff}}$ can change sign this is ill-posed without some additional cutoff for short wavelengths (on the $Y$ scale). Incorporation of a cutoff would not be unreasonable given that the theory is based on scale separation and a threshold can be determined in related models that allow zonostrophic instability on arbitrary scales (Srinivasan & Young Reference Srinivasan and Young2012). For example, negative effective viscosity at large scale can be cut off by positive effective hyperviscosity at smaller scales (and Sukoriansky et al. (Reference Sukoriansky, Galperin and Chekhlov1999) argue that in two-dimensional turbulence these effects also need to be time-dependent to simulate subgrid-scale motion). We defer these considerations to future study, and here note that although we cannot integrate (3.28) explicitly, we can seek equilibrium profiles, that is, mean field profiles independent of the time scale $\mathscr{T}$ . Integrating (3.28) once, we can write

with a constant $C$ . To plot solutions it is convenient to reduce the number of parameters by setting

so that we solve

to obtain a family of solutions depending on only one parameter  $\tilde{{\it\lambda}}$ .

Now we numerically integrate (3.36) over a range of ${\tilde{Y}}$ with $\tilde{{\it\Omega}}$ passing through $\tilde{{\it\Omega}}=0$ ; this gives a family of profiles depicted in figure 4(a) for $\tilde{{\it\Omega}}$ and (b) for $\tilde{U}$ . It turns out that we cannot obtain sensible solutions unless the effective viscosity $[\cdots ]$ in (3.36) remains positive, and this restricts $\tilde{{\it\lambda}}^{-3}<2$ . This is equivalent to being below the threshold for zonostrophic instability (without background uniform shear) in (3.33). As $\tilde{{\it\lambda}}^{-3}$ is made closer to 2, we have a lower effective viscosity in a region close to $\tilde{{\it\Omega}}=0$ , and so the scope to sustain larger gradients of $\tilde{{\it\Omega}}$ . This is seen in figure 4(a), while (b) shows the increasing sharpness of the $\tilde{U}$ profile and (c) shows the corresponding dip in the total viscosity, i.e.  ${\it\lambda}+{\it\nu}_{\mathit{eff}}$ , to a small but positive value.

Using (3.29) we can also obtain the approximate form of the $\tilde{{\it\Omega}}$ profile as an inverse cubic,

and in the limit $\hat{{\it\lambda}}^{-3}\rightarrow 2$ we obtain profiles

(up to appropriate additive constants).

Figure 4.

Plots of equilibrium profiles found by integrating the ODEs (3.36). Here, $2-\tilde{{\it\lambda}}^{-3}=1$ , $1/2$ , $1/4,\ldots ,1/16$ , and as $\tilde{{\it\lambda}}^{-3}$ approaches the value 2, smaller-scale structure is evident in the curves for (a)  $\tilde{{\it\Omega}}({\tilde{Y}})$ (reading up the curves), (b)  $\tilde{U} ({\tilde{Y}})$ (down) and (c)  $\tilde{{\it\lambda}}^{-3}\tilde{{\it\Omega}}^{-2}\,h_{\mathit{R}}(\tilde{{\it\Omega}}^{-1})+1$ (down, on the left-hand side).

4.1 Multiple-scale formulation

We follow the discussion at the beginning of § 3 and set ${\it\sigma}(y,t)=\text{e}^{-\text{i}mU(y)t}\,c(y,t)$ . With the scalings

(and $Y$ , $T$ as before), we obtain from (2.13)

for the waves and

for the mean field, from (2.22). The scalings for the fluxes in (4.2) differ from those earlier in (3.8) because

does not have the same leading-order cancellation as that seen for $F_{\mathit{R}}$ , discussed below (3.5). Although the equations for vorticity and a passive scalar have similarities, this is one key difference, the other being that the vorticity equation includes an explicit source term, the external forcing, where for a passive scalar there is no source, only a background gradient.

The leading-order passive scalar problem is easily integrated, using (3.23),

(with $D$ defined in (3.22)), and we may write the integrated flux as

with $h_{\mathit{P}}=h_{\mathit{P}}(\hat{{\it\lambda}},\hat{{\it\beta}})$ from (3.27). This appears in the large-scale passive scalar transport equation

Figure 5.

Plot of (a) $f_{\mathit{P}}$ as a function of ${\it\tau}$ and (b)  $h_{\mathit{P}}$ as a function of  $\hat{{\it\lambda}}$ . In each plot the values are $\hat{{\it\beta}}=0,1,\ldots ,5$ , reading down the curves on the left.

As in the previous section, with similar notation, the function $f_{\mathit{P}}$ gives the instantaneous flux of the passive scalar (from a uniform background) in the presence of a wave initiated by the external forcing at $T=0$ . Unlike in the previous section, $f_{\mathit{P}}$ depends on the parameter $\hat{{\it\beta}}$ as well as the rescaled time ${\it\tau}$ , but not on  $\hat{{\it\lambda}}$ . Figure 5(a) shows $f_{P}$ for varying values of $\hat{{\it\beta}}$ ; as this parameter is increased the profile becomes increasingly oscillatory and the integrated transport is reduced. This is shown in figure 5(b) depicting $h_{\mathit{P}}$ as a function of $\hat{{\it\lambda}}$ for various values of $\hat{{\it\beta}}$ , showing the suppression of transport as the ${\it\beta}$ -effect is increased (see, e.g., Srinivasan & Young Reference Srinivasan and Young2014) and the flows become more wave-like in nature. We also note that

4.2 Equilibrium profiles

Although in figure 5 we see variation in the transport of the passive scalar as $\hat{{\it\lambda}}$ and $\hat{{\it\beta}}$ are varied, the effect on the equilibrium passive scalar profiles turns out not to be as dramatic as for the vorticity transport in § 3.5. The steady-state version of (4.9) is

where the constant $C_{{\it\Sigma}}$ would be fixed by the overall gradient of the passive scalar across the system. We rescale

together with (3.35), so that we can write from (4.11)

To obtain a passive scalar profile we first fix $\tilde{{\it\lambda}}^{-3}=2-1/16$ , giving the sharpest $\tilde{{\it\Omega}}$ profile depicted in figure 4(a). We then integrate (4.13) for any value of $\tilde{{\it\beta}}$ , and the resulting values of $\tilde{{\it\Sigma}}({\tilde{Y}})$ are shown in figure 6(a) for $\tilde{{\it\beta}}=0,1,\ldots ,5$ , reading up the curves. There is little effect on sharpening the passive scalar profiles in (a), although the plots of the gradient $\partial _{{\tilde{Y}}}\tilde{{\it\Sigma}}$ in (b) show more structure. It should be noted that if the passive scalar is the vector potential of a kinematic magnetic field, then figure 5(b) represents the field itself, reduced to approximately half the exterior value within the step for ${\it\beta}=0$ . This may be thought of as a process of flux expulsion (Weiss Reference Weiss1966), from regions of stronger motion corresponding to greater effective scalar diffusivity.

Figure 6.

Plots of equilibrium scalar profiles by integrating the ODE (4.13). Here, $2-\tilde{{\it\lambda}}^{-3}=1/16$ . For values of $\tilde{{\it\beta}}=0,1,\ldots ,5$ , in (a)  $\tilde{{\it\Sigma}}({\tilde{Y}})$ (reading up the curves) and in (b)  $\partial _{{\tilde{Y}}}\tilde{{\it\Sigma}}$ (reading up) are plotted against ${\tilde{Y}}$ .

Somewhat paradoxically then, a step jump can be supported in the vorticity profile of figure 4(a), whereas a passive scalar profile remains relatively unperturbed. Of course, different types of transport are involved, and whereas the effective viscosity can become negative and nearly cancel out the molecular viscosity to allow a jump, the same is not true for the passive scalar (or vector potential). Further investigations for the case of different values of the scalar diffusivity and viscosity in (2.7) would be of interest, but the governing ODEs would need to be solved numerically.

5.1 Multiple-scale formulation

We again remove the fast advection terms in (2.11), (2.12), by setting

(as listed in table 1), to obtain

with corresponding fluxes $F_{\mathit{R}}$ in (3.5),

For the large-scale expansion we replace

As before, we adopt $Y={\it\varepsilon}y$ , $T={\it\varepsilon}t$ , $\mathscr{T}={\it\varepsilon}^{3}t$ , as well as the replacements

referring to the small-scale fields, and

for the fluxes. This yields, exactly,

with the large-scale equations

recalling (2.23). The flux $F_{\mathit{R}}$ is given in (3.11); we gain a similar term from the Lorentz force,

and $F_{\mathit{M}}$ remains as in (5.6).

5.2 Numerical solution

Again, before embarking on analysis, we present a simulation of the full wave problem with ${\it\beta}=0$ , ${\it\varepsilon}=0.1$ , $U=\cos Y$ and a constant field $B=2$ . The exact rescaled equations are

(The fluxes $F_{\mathit{R}}$ , $F_{\mathit{L}}$ are rescaled as in (3.17) whereas $F_{\mathit{P}}$ is not.) Figure 7 shows the real parts of the fields ${\it\omega},{\it\psi},j,a$ as functions of $(T,Y)$ in (a–d), with the corresponding fluxes in (e–h). As well as the general decrease in scale as $T$ increases, one can clearly see the presence of oscillations, Alfvén waves, in which energy is transferred between the fluid flow and the magnetic field in (a–d). With these are oscillations in the instantaneous fluxes giving the feedback on the mean flow and field.

Figure 7.

Simulation of waves with ${\it\beta}=0$ , $m=1$ , ${\it\varepsilon}=0.1$ , ${\it\lambda}=0$ , $U=\cos Y$ and $B=2$ . In (a)  $\text{Re}\,{\it\omega}$ , (b)  $\text{Re}\,{\it\psi}$ , (c)  $\text{Re}\,j$ , (d)  $\text{Re}\,a$ , (e)  $F_{\mathit{R}}$ , (f)  $F_{\mathit{L}}$ , (g)  $F_{\mathit{K}}$ and (h)  $F_{\mathit{M}}$ are plotted in the $(T,Y)$ -plane.

5.3 Multiple-scale expansion

To analyse the complicated picture emerging in figure 7, we again expand all small-scale fields in powers of ${\it\varepsilon}$ to yield the leading-order wave equations, from (5.10), (5.11),

These are to be solved with initial conditions from (2.15) and the results fed into the mean equations via fluxes (dropping the terms of order  ${\it\varepsilon}$ ),

Unfortunately, the above ODEs (5.18)–(5.20) are not soluble analytically in simple terms (that is, without recourse to so-called Heun functions). They can be solved approximately in the limit of strong and weak shear as measured by $\hat{B}$ below (Leprovost & Kim Reference Leprovost and Kim2009) to obtain expressions for the effective transport coefficients for the case of isotropic driving.

In the first instance we wish to proceed without approximation. We first rescale and remove the diffusive decay by setting

with the damping term $D$ given in (3.22), to leave equations for sheared Alfvén waves,

Although we cannot solve the ODEs, we proceed for the moment as if we knew the solution with all quantities expressed as functions of $(T,{\it\Omega},B,{\it\beta})$ . We would then evaluate the fluxes by differentiating these expressions, and could write the instantaneous fluxes in the form

It should be noted that, crucially, the fluxes depend on $Y$ not only through gradients of the vorticity profile ${\it\Omega}^{\prime }$ (as in the hydrodynamic problem earlier) but also through the gradients $B^{\prime }$ in the magnetic field profile. This feature, discussed by Chechkin (Reference Chechkin1999), Kim (Reference Kim2007) and Leprovost & Kim (Reference Leprovost and Kim2009), creates additional complexity in the MHD problem.

Following through with similar notation to § 3.3, the functions $f_{\mathit{Z}}$ encapsulate all of the interesting time evolution of the fluxes, and this may be most conveniently expressed by first extending the definitions in (3.22) to include

We may then write all of the functions in terms of ${\it\tau}$ and three parameters $\hat{B}$ , $\hat{{\it\beta}}$ and $\hat{{\it\lambda}}$ , with

Here, the functions $f_{{\it\Omega}\mathit{R}}$ and $f_{{\it\Omega}\mathit{L}}$ capture the effect of a vorticity gradient ${\it\Omega}^{\prime }$ on the Reynolds stress and Lorentz force, while $f_{B\mathit{R}}$ and $f_{B\mathit{L}}$ give the effect in the Navier–Stokes equation of a gradient of background magnetic field $B^{\prime }$ . The transport of scalar vector potential $A$ is linked to $B=A^{\prime }$ through the $f_{\mathit{M}}$ term.

We now extend the convention in (2.23), so that $\mathit{Z}_{\mathit{K}}=\mathit{Z}_{\mathit{R}}-\mathit{Z}_{\mathit{L}}$ for any quantity $\mathit{Z}$ , and express the integrated fluxes in terms of transport coefficients ${\it\nu}_{\mathit{eff}}$ , ${\it\chi}_{\mathit{eff}}$ and ${\it\eta}_{\mathit{eff}}$ ,

This leads to coupled large-scale equations

with (making use of (3.27))

We have reached the end of an analytical development for the MHD problem mirroring that for the hydrodynamic framework in § 3.3. If we could solve the ODEs (5.23) explicitly to obtain $\hat{{\it\zeta}}_{0},\hat{{\it\phi}}_{0},\hat{{\it\alpha}}_{0},\hat{{\it\gamma}}_{0}$ , taken as functions of $(T,{\it\Omega},B,{\it\beta})$ , we would then calculate the functions $f_{\mathit{Z}}$ in (5.28)–(5.32), differentiating quantities as needed. Although we cannot solve these ODEs analytically in general, we can do so numerically, and in fact we can add to these ODEs further equations for derivatives such as ${\it\Omega}\partial _{{\it\Omega}}|\hat{{\it\zeta}}_{0}|^{2}$ or $B\partial _{B}|\hat{{\it\zeta}}_{0}|^{2}$ . We relegate these uninspiring details to (B 1)–(B 6) in appendix B.

Figure 8.

Plots of $f_{{\it\Omega}\mathit{R}}$ (solid), $f_{{\it\Omega}\mathit{L}}$ (dashed) and $f_{{\it\Omega}\mathit{K}}$ (dotted) as functions of ${\it\tau}$ for $\hat{{\it\lambda}}=\hat{{\it\beta}}=0$ and (a)  $\hat{B}=0.5$ and (b)  $\hat{B}=2.0$ .

Figure 9.

Plots of $f_{B\mathit{R}}$ (solid), $f_{B\mathit{L}}$ (dashed) and $f_{B\mathit{K}}$ (dotted) as functions of ${\it\tau}$ for $\hat{{\it\lambda}}=\hat{{\it\beta}}=0$ and (a)  $\hat{B}=0.5$ and (b)  $\hat{B}=2.0$ .

Figure 10.

Plots of $f_{M}$ as a function of ${\it\tau}$ for $\hat{{\it\lambda}}=\hat{{\it\beta}}=0$ and (a)  $\hat{B}=0.5$ and (b)  $\hat{B}=2.0$ .

With this set-up, figures 8–10 show the transport terms $f_{\mathit{Z}}$ as functions of ${\it\tau}$ for $\hat{{\it\lambda}},\hat{{\it\beta}}=0$ and values $\hat{B}=0.5$ and 2. The results for $f_{{\it\Omega}\mathit{R}}$ (solid) in figure 8 may be compared with $f_{\mathit{R}}$ (outer curve) in figure 2(a); it is clear how increasing $\hat{B}$ gives rise to oscillations in these fluxes because of Alfvén waves. Likewise, $f_{\mathit{M}}$ in figure 10 may be compared with $f_{\mathit{P}}$ for a passive scalar in figure 5(a) (outer curve).

Figure 11.

Plots of (a)  $\hat{{\it\lambda}}^{2}h_{{\it\Omega}\mathit{R}}$ , (b)  $\hat{{\it\lambda}}^{2}h_{{\it\Omega}\mathit{L}}$ , (c)  $\hat{{\it\lambda}}^{2}h_{{\it\Omega}\mathit{K}}$ , (d)  $\hat{{\it\lambda}}^{2}h_{B\mathit{R}}$ , (e) $\hat{{\it\lambda}}^{2}h_{B\mathit{L}}$ , (f)  $\hat{{\it\lambda}}^{2}h_{B\mathit{K}}$ , (g)  $\hat{{\it\lambda}}^{2}h_{\mathit{M}}$ , (h)  $\hat{{\it\lambda}}^{2}m^{3}{\it\Omega}B({\it\eta}_{\mathit{eff}}+B\,\partial _{B}{\it\eta}_{\mathit{eff}})$ (see (B 11)) and (i)  $\hat{{\it\lambda}}^{2}m^{3}{\it\Omega}^{2}\,\partial _{{\it\Omega}}{\it\eta}_{\mathit{eff}}$ (see (B 12)) as functions of $\hat{{\it\lambda}}$ for $\hat{B}=0$ , 1, 2, 3, 4 (solid) and 5 (dashed), with $\hat{{\it\beta}}=0$ .

We now begin to suffer from the presence of too many parameters in the problem and struggle to plot the dependence of quantities $f_{\mathit{Z}}({\it\tau},\hat{B},\hat{{\it\beta}},\hat{{\it\lambda}})$ . However, the quantities $\hat{{\it\lambda}}^{2}h_{\mathit{Z}}(\hat{B},\hat{{\it\beta}},\hat{{\it\lambda}})$ (the prefactor chosen to show the clearest structure) are plotted as functions of $\hat{{\it\lambda}}$ for $\hat{{\it\beta}}=0$ and different values of $\hat{B}$ in figure 11. The values are $\hat{B}=0$ , $1,\ldots ,5$ , with the last curve dashed to mark it out. Where there are straight lines, this reflects the lack of coupling in the non-magnetic case $\hat{B}=0$ . Some effects can clearly be seen, for example the effect of magnetic field is to suppress transport of $A$ , via suppression of ${\it\eta}_{\mathit{eff}}$ or $h_{\mathit{M}}$ , depicted in figure 11(g). These curves have been carefully confirmed against analytical approximations (see appendix B) for varying values of $\hat{B},\hat{{\it\lambda}},\hat{{\it\beta}}\gg 1$ as a partial check on our analysis and coding.

5.4 Zonostrophic instability

Although any calculations become somewhat opaque at this point, as we can only evaluate many quantities numerically, we can still press on to study zonostrophic instability for the MHD system, based on the large-scale equations (5.34), (5.35). These are complicated as we have three effective transport terms, which themselves depend on the local vorticity ${\it\Omega}$ and magnetic field  $B$ .

We therefore think of the coefficients ${\it\nu}_{\mathit{eff}},{\it\chi}_{\mathit{eff}},{\it\eta}_{\mathit{eff}}$ as functions of $({\it\Omega},B,{\it\beta},{\it\lambda})$ and linearise (5.34), (5.35) about a state of constant vorticity ${\it\Omega}_{0}$ and field $B_{0}$ . We thus replace

where now ${\it\Omega}$ , $B=A^{\prime }$ and $A$ are small quantities. We then obtain

where the coefficients and their derivatives are evaluated at $({\it\Omega},B)=({\it\Omega}_{0},B_{0})$ . In general, the equations are completely coupled, and if we seek a mode with $({\it\Omega},B)$ proportional to $\exp (PK^{2}\mathscr{T}+\text{i}KY)$ we find growth rates $P$ satisfying

where $\unicode[STIX]{x1D644}$ is the identity matrix.

We wish to investigate instabilities as a function of $({\it\Omega}_{0},B_{0},{\it\beta},{\it\lambda})$ , and we commence by setting ${\it\beta}=0$ and ${\it\Omega}_{0}=0$ , no background shear. This is a major simplification, because for given ${\it\lambda}$ , $B_{0}$ , we use the limiting expressions for ${\it\Omega}_{0}\rightarrow 0$ , that is for $\hat{{\it\lambda}},\hat{B}\gg 1$ . We are again effectively in a viscous regime (on the ${\it\tau}$ time scale) in which the exponential damping effect of the diffusion terms is dominant, for ${\it\tau}=O(\hat{{\it\lambda}}^{-1})\ll 1$ . On this time scale $\hat{B}{\it\tau}=O(1)$ also and so we need to use

as leading-order approximations to solutions of (5.23) (see appendix B). (It should be noted that a straightforward Maclaurin expansion in powers of ${\it\tau}$ is not satisfactory as this series becomes non-uniform when $\hat{B}{\it\tau}=O(1)$ ; our series is valid provided only that $\hat{B}{\it\tau}^{2}\ll 1$ .) This gives expressions for the $h_{\mathit{Z}}$ that we do not set out here, but that have been used to check the numerical calculations in figure 11. More interestingly, we obtain

valid for fixed ${\it\lambda}$ and $B$ with ${\it\Omega}\rightarrow 0$ , at leading order.

Figure 12.

Contour plot of $m^{2}P$ (5.43) in the $(m^{2}{\it\lambda},mB_{0})$ -plane for ${\it\Omega}_{0}={\it\beta}=0$ . Contours increase from $m^{2}P=0$ (outermost) in steps of 0.1 to $m^{2}P=10$ .

Turning to (5.40), at ${\it\Omega}={\it\Omega}_{0}=0$ both diagonal entries vanish, and in addition for the bottom right entry ${\it\eta}_{\mathit{eff}}+B\partial _{B}{\it\eta}_{\mathit{eff}}=-{\it\nu}_{\mathit{eff}}$ at this order. Thus, we are left with explicit expressions for the growth rates,

The corresponding growth rate contours for $m^{2}P\geqslant 0$ are depicted in figure 12. The lower set of contours, intersecting the horizontal axis, are for the first set of eigenvalues in (5.43) and correspond to the hydrodynamic zonostrophic instability in § 3.4, modified by the magnetic field. This has a suppressing effect, switching off the instability for $mB_{0}$ larger than approximately 0.34, in at least qualitative agreement with the results of Tobias et al. (Reference Tobias, Hughes and Diamond2012).

The second set of eigenvalues in (5.43) gives contours intersecting the vertical axis in figure 12. This is a magnetically driven instability, in which the variations of diffusivity ${\it\eta}_{\mathit{eff}}$ with $B$ lead to the tendency of the field to segregate into regions of weaker and stronger $B$ . The effect of removing the field from regions of overturning motion is known as flux expulsion (Weiss Reference Weiss1966); in such regions the Lorentz force feedback is then reduced, increasing the fluid flows and so enhancing the flux expulsion, giving a runaway effect. The zonostrophic instability is restricted to a range of ${\it\lambda}$ values for $B_{0}=0$ , whereas the magnetic flux expulsion instability occurs for any $B_{0}$ as ${\it\lambda}\rightarrow 0$ . It should be noted that this latter instability does not correspond to a sign change in ${\it\chi}_{\mathit{eff}}$ , which would be impossible as the effective diffusivity arises from advection–diffusion of $A$ , but is seen in (5.40) to emerge from the variation of ${\it\chi}_{\mathit{eff}}$ with respect to changes in the magnetic field  $B$ .

Figure 13.

Contour plot of $m^{2}P$ from (5.40) in the $(m^{2}{\it\lambda},mB_{0})$ -plane for ${\it\beta}=0$ and (a,b)  ${\it\Omega}_{0}=0.1$ and (c,d)  ${\it\Omega}_{0}=0.2$ . In (a,c) the real part of $m^{2}P$ is shown and in (b,d) the imaginary part. Contours increase from $m^{2}P=0$ (outermost) in steps of 0.1 to 10.

From this calculation we can now include a background shear, linearising about $({\it\Omega}_{0},B_{0})$ with ${\it\Omega}_{0}>0$ . As in § 3.4, we can no longer evaluate the quantities ${\it\nu}_{\mathit{eff}}$ , ${\it\chi}_{\mathit{eff}}$ and ${\it\eta}_{\mathit{eff}}$ analytically to obtain values of $P$ from (5.40), and so we compute them numerically. Figure 13 shows the contour plot of $m^{2}P$ for (a,b) ${\it\Omega}_{0}=0.1$ and (c,d) 0.2. The real parts of $m^{2}P$ are shown in (a,c), and these reveal that shear tends to suppress both types of instability. Interestingly though, there emerges near to the line $mB_{0}=m^{2}{\it\lambda}$ a region of oscillatory instability, with a pair of complex conjugate values of $m^{2}P$ , whose imaginary parts are depicted in (b,d). This suggests emergence of a coupled instability through overstability. This can be seen to some extent in our analysis: in (5.40), the product ${\it\chi}_{\mathit{eff}}B_{0}\partial _{{\it\Omega}}{\it\eta}_{\mathit{eff}}$ is negative (near $mB_{0}=m^{2}{\it\lambda}$ ) in our analytical approximation, from (5.42), (B 10). It also corresponds to the product of the quantities shown in figure 11(f,i) being negative, at least for small  $B$ . Therefore, if the actual diagonal terms are small, the off-diagonal terms can give an imaginary part in $m^{2}P$ . The oscillatory eigenvalue could be considered as resulting from a resonance between the two instabilities in figure 12.

Figure 14.

Contour plot of $m^{2}P$ (3.32) in the $(m^{2}{\it\lambda},mB_{0})$ -plane for ${\it\Omega}_{0}=0$ and (a,b)  ${\it\beta}=0.2$ and (c,d)  ${\it\beta}=0.4$ . In (a,c) contours for the zonostrophic instability are shown and in (b,d) contours of the magnetic segregation instability. Contours increase from $m^{2}P=0$ (outermost) in steps of 0.1 to 10.

The other effect we can include is that of a background vorticity gradient, ${\it\beta}\neq 0$ . We turn off the background shear, ${\it\Omega}_{0}=0$ , and reintroduce the parameter $\hat{{\it\beta}}$ . The corresponding transport coefficients may be computed, as we are again in the limit $\hat{{\it\lambda}},\hat{{\it\beta}},\hat{B}\gg 1$ as ${\it\Omega}_{0}\rightarrow 0$ , and these are given in (B 14)–(B 16). The off-diagonal elements are again zero (at this order) in the matrix (5.40), and so the problem decouples to eigenvalues given by the diagonal entries only. Figure 14 shows plots of the zonostrophic modes in (a,c) and the magnetic flux expulsion modes in (b,d) for ${\it\beta}=0.2$ and 0.4. The effect of increasing ${\it\beta}$ is to suppress the magnetic instability, panels (b,d). However, the zonostrophic instability contours now embrace the vertical axis. The combined effect of the magnetic field $B_{0}$ and ${\it\beta}$ is to enhance the zonostrophic instability, particularly for small viscosity. This can be contrasted with figure 12 for ${\it\beta}=0$ , where the zonostrophic contours remain below the line $m^{2}{\it\lambda}=mB_{0}$ . There is supporting evidence of enhanced instability in full numerical simulations (Durston Reference Durston2015).

Finally, we note that the effective terms ${\it\nu}_{\mathit{eff}}$ and ${\it\chi}_{\mathit{eff}}$ include contributions from both the Reynolds stress and Lorentz force terms. These contributions can be expressed as

Thus, whereas two terms contribute equally and additively to ${\it\chi}_{\mathit{eff}}$ , for the effective viscosity there is a leading-order cancellation for large magnetic field $B\gg m{\it\lambda}$ . The effect of the Lorentz force in cancelling Reynolds stresses for strong magnetic fields, and the overall suppression of transport for large $B$ evident in (5.42) are discussed in Leprovost & Kim (Reference Leprovost and Kim2009).

5.5 Equilibrium profiles

Finally, we consider steady equilibrium profiles in the full MHD system. For a steady profile we have

In addition to (3.35), (4.12), we rescale

here, $A$ is analogous to the passive scalar in (4.12). Making use of (5.36), we obtain the equations

with now the functions

which we have to evaluate numerically. It should be noted that here $\tilde{B}(Y)$ gives the magnetic field profile, but $\tilde{B}_{0}$ is a constant, a measure of the average field strength in the system. Given values for the parameters $\tilde{{\it\lambda}}$ , $\tilde{{\it\beta}}$ and $\tilde{B}_{0}$ , integrating the differential algebraic system (5.49)–(5.51) determines profiles for the fields.

Figure 15.

Plots of equilibrium profiles by integrating the ODEs (5.49)–(5.51) with $2-\tilde{{\it\lambda}}^{-3}=1/16$ . Panel (a) shows $\tilde{{\it\Omega}}$ (reading down the curves) and (b) shows $\tilde{B}$ (reading up on the left side) for $\tilde{{\it\beta}}=0$ and $\tilde{B}_{0}=0,0.2,\ldots ,1$ . Panel (c) shows $\tilde{{\it\Omega}}$ (reading up) and (d) shows $\tilde{B}$ (reading up) for $\tilde{B}_{0}=1$ and $\tilde{{\it\beta}}=0,1,\ldots ,5$ .

We illustrate some profiles in figure 15. It should be noted that in the limit $\tilde{B}_{0}=0$ , we regain the decoupled hydrodynamic and passive scalar problem, with profiles shown in figures 4(a) and 6(b). With this in mind, in figure 15(a,b) we show the profiles $\tilde{{\it\Omega}}({\tilde{Y}})$ and $\tilde{B}({\tilde{Y}})$ respectively, where we start with $\tilde{B}_{0}=0$ and $\hat{{\it\lambda}}^{-3}=2-(1/16)$ (close to the threshold for zonostrophic instability). This gives the sharpest profiles for $\tilde{{\it\Omega}}$ and $\tilde{B}$ as in the earlier figures, and as the parameter $\tilde{B}_{0}$ is increased we observe smoother curves: the step in vorticity and the effect of flux expulsion are smoothed out in the presence of increasing magnetic fields. This could also be viewed as the effect of increasing magnetic field in moving the point in parameter space away from the threshold for zonostrophic instability (see figure 12), and indeed if $\hat{{\it\lambda}}$ is also adjusted, the profiles can be steepened again for non-zero magnetic field (not shown).

We continue to fix $\hat{{\it\lambda}}^{-3}=2-(1/16)$ , take $\tilde{B}_{0}=1$ giving the two sharpest profiles in figure 15(a,b), and then consider the effects of increasing $\tilde{{\it\beta}}$ in figure 15(c,d). Interestingly, the effect of increasing $\tilde{{\it\beta}}$ is to sharpen the vorticity profile again, while the magnetic field profile becomes smoother. The behaviour of the magnetic field profile is very similar to that of the analogous passive scalar gradient in figure 6(b). However, the sharpening of the vorticity profile is not observed in the purely hydrodynamic system, and in that case there is actually no dependence on $\tilde{{\it\beta}}$ for vorticity profiles. Therefore, we observe that the magnetic field has a catalysing effect: its presence allows the ${\it\beta}$ -effect to sharpen the vorticity profile. This is in line with the earlier discussion of zonostrophic instabilities: the combination of ${\it\beta}$ -effect and magnetic field can be destabilising, leading to the increased areas of zonostrophic instability in figure 14(a,c).