3.1. Prediction

The symmetry of the initial momentum profile $M(y)$ about the equator ( $y=0$ ) and the fact that equal rates of initial growth of the instability will occur in the two instability regions, suggests mixing will occur between $y=\pm y_{mix}$ with $m_{c}=M(y_{mix})$ such that $\int (M(y)-m_{c})\text{d}y=0$ . This is sketched in figure 3(a). The corresponding new flow would be stable because all negative gradients $\text{d}m/\text{d}y$ would have disappeared for $y<0$ and all positive gradients for $y>0$ .

Figure 3.

(a) Schematic of the predicted momentum distribution. The thin curve is the initial $m$ -distribution  $M(y)$ for an unstable Gaussian jet ( $\mathit{Ro}<-1/2$ ). Instability commences in the two regions, on the left $y\in [\,y_{-},\,0]$ which connects to the region on the right with $y\in [0,\,y_{+}]$ as also indicated in figure 2(b). The values of $y_{\pm }$ are a function of the Rossby number according to (3.7). The left instability region is associated with $\text{d}M/\text{d}y<0$ and right instability region with $\text{d}M/\text{d}y>0$ so that $y\text{d}M/\text{d}y>0$ or equivalently ${\it\Phi}<0$ . The outer limits $\{y_{-},y_{+}\}$ are indicated with the thin dashed vertical lines. The prediction for the equilibrated flow is indicated with the thick solid curve. It has constant $m=m_{c}<0$ between $y=\pm y_{mix}$ indicated by the thin solid vertical lines. Equation (3.9) determines how non-dimensional $m_{c}$ and $y_{mix}$ vary with the Rossby number when $\mathit{Ro}<-1/2$ (see text). (b) The initial Gaussian jet (thin line) and the predicted final flows according to (3.10) for $\mathit{Ro}=-2$ and $\mathit{Ro}=-4$ . In each case the amplitudes have been scaled with $|\mathit{Ro}|$ to facilitate comparison. For $\mathit{Ro}=-2$ the maximum amplitude (at  $y=0$ ) has decreased 25 %, for $\mathit{Ro}=-4$ the decrease is 40 %.

If this is indeed what happens, then $y_{mix}$ is determined by the condition

(3.8) $$\begin{eqnarray}\int _{-y_{mix}}^{+y_{mix}}\{|\mathit{Ro}|\exp (-y^{2})+y^{2}/2\}\text{d}y=\int _{-y_{mix}}^{+y_{mix}}\{|\mathit{Ro}|\exp (-y_{mix}^{2})+y_{mix}^{2}/2\}\text{d}y,\end{eqnarray}$$

which yields the transcendental relation between $y_{mix}$ and $|\mathit{Ro}|$

(3.9) $$\begin{eqnarray}\int _{0}^{+y_{mix}}|\mathit{Ro}|\exp (-y^{2})\text{d}y=y_{mix}|\mathit{Ro}|\exp (-y_{mix}^{2})+y_{mix}^{3}/3,\end{eqnarray}$$

which is easily solved numerically. Knowing $y_{mix}$ as a function of $|\mathit{Ro}|$ we also know $m_{c}(|\mathit{Ro}|)$ , and we can predict that the adjusted absolute momentum and velocity profile non-dimensionally are

(3.10) $$\begin{eqnarray}\left.\begin{array}{@{}lll@{}}m=m_{c}, & u=m_{c}+y^{2}/2, & y\in [-y_{mix},+y_{mix}],\\ m=M(y), & u=U(y), & y\notin [-y_{mix},+y_{mix}],\end{array}\right\}\end{eqnarray}$$

with non-dimensional $U$ as in (3.2). Note that for all negative $\mathit{Ro}$ , according to (3.2), the initial $m(y)<0$ for all $y$ and that for increasing $|\mathit{Ro}|$ the value of the predicted $m_{c}$ gets ever more negative while the predicted mixing region $[-y_{mix},+y_{mix}]$ gets ever broader. This is illustrated in figure 3(b) where we plot the velocity profiles we expect to find after equilibration of the inertial instability for $\mathit{Ro}=-2$ and $\mathit{Ro}=-4$ .

3.2. Numerical simulations

We now demonstrate that numerical simulations confirm the predictions for the equilibration of the westward flowing Gaussian jet at the equator. In figure 4, we show the evolution of the vertically averaged profiles of $m$ and $u$ as the inertial instability unfolds and equilibrates. As discussed in § 2, the vertical wavelength of the inertial instability vanishes at infinite Reynolds number. All of the simulations that we present will have a finite Reynolds number, which we define in the Gaussian case as $\mathit{Re}=|U_{0}|L/{\it\nu}$ . With viscosity, the most unstable mode has a finite vertical wavelength. The resolution that we use in all the simulations presented here will be sufficient to resolve the most unstable mode and also, as best as we can, all of the significant smaller scale motions that are generated nonlinearly. In the simulation used to prepare figure 4, $\mathit{Re}=15$ k (i.e. 15 000). The Rossby number is as defined in (3.2), and in this simulation $\mathit{Ro}=-4$ .

Figure 4.

Profiles of vertically averages of (a) absolute momentum $m$ and (b) zonal velocity $u$ for a westward flowing Gaussian zonal jet centred on the equator. Here $\mathit{Ro}=-4$ and $\mathit{Re}=|U_{0}|L/{\it\nu}=15$ k. The full computational domain for this simulation is such that $z\in [0,4{\rm\pi}]$ and $y\in [-6,+6]$ on a grid of $513\times 513$ points. Here we are showing only the $y\in [-3,+3]$ portions of the profiles. Thick grey lines indicate the prediction. Along the axis $y$ is non-dimensionalized with the length scale $L$ . Time has been non-dimensionalized with the time scale $({\it\beta}L)^{-1}$ .

In figure 4(a), we see that the profile of vertically averaged momentum changes rapidly from the initial condition (thin grey line) to a nearly equilibrated state at non-dimensional time $t=18$ . The predicted equilibrium is shown as a thick grey line. After $t=18$ , the evolution is much slower, with the profile becoming smoother as seen at $t=50$ . The subsequent evolution is primarily slow viscous diffusion of momentum. The $m$ -profile evolves toward the homogeneous mixed state in the mixing region as predicted by our mixing scheme. The corresponding evolution of the profile of vertically averaged velocity $u$ is shown in figure 4(b). The profile rapidly evolves toward the predicted profile. As predicted, the maximum velocity of the jet is reduced in value by about a factor of a half during the equilibration. The match between the profile at $t=50$ (thin solid black curve) and the prediction (thick grey curve) is excellent for most of the jet. In the region just outside of the predicted mixing region, there is a small overshoot at $t=50$ in which a small amount of eastward flow is found. This has also been noted before in examples of flow on the $f$ -plane (see Kloosterziel et al. Reference Kloosterziel, Orlandi and Carnevale2007b ).

Figure 5 shows how the results of equilibration vary with $\mathit{Re}$ and $\mathit{Ro}$ . Figure 5(a) confirms that, as in our previous work with flows on the $f$ -plane, the match between the vertically averaged profile of $m$ and the prediction improves with increasing $\mathit{Re}$ . Here the view is ‘zoomed-in’ relative to that shown in figure 4(a), and we see that the curve for $m$ at $t=50$ and $\mathit{Re}=15$ k is not perfectly flat. The irregularities are the result of perturbation vortices (see below) that have not yet dissipated. Figure 5(b) shows that equilibration occurs similarly over a wide range of unstable $\mathit{Ro}$ .

Figure 5.

Effects of variation of (a)  $\mathit{Re}$ and (b)  $\mathit{Ro}$ on the equilibration of inertial instability of the Gaussian jet. All of the simulation results are taken at $t=50$ . The size of the computational domain and the resolution are as described in the caption of figure 4. Thick grey lines indicate the predictions.

The vortex dynamics of the equilibration process for the Gaussian zonal jet on the equator in the standard ${\it\beta}$ -plane model is illustrated in figure 6. Only the along-stream component of the vorticity ${\it\omega}_{x}=\partial w/\partial y-\partial v/\partial z$ is shown since it is the growth of this component that characterizes the inertial instability, that is, the inertial instability is characterized by secondary overturning motions with their axis aligned with the flow direction as sketched in figure 1(a).

Figure 6.

Contour plots of ${\it\omega}_{x}$ from a simulation of the Gaussian jet with $\mathit{Ro}=-4$ and $\mathit{Re}=15$ k at non-dimensional times (a)  $t=7.5$ , (b) 10, (c) 20, (d) 50. The along-flow component of the vorticity is ${\it\omega}_{x}=\partial w/\partial y-\partial v/\partial z$ with $v,w$ the velocity components in the $y$ and $z$ direction, respectively. The initial instability region, is bounded by the vertical dashed lines, at $y_{-}$ and $y_{+}$ . The predicted outer limits of the inertial instability equilibration range, $y=-y_{mix}$ and $y=+y_{mix}$ , are indicated by the vertical solid black lines. Positive/negative contours are drawn as black/grey lines. The zero contour level is not drawn. The contour increments are (a) 0.5, (b) 4.0, (c) 4.0, (d) 1. In (a), the vertical solid grey lines indicate the initial positions $\pm y_{max}$ of maximum growth rate according to linear theory. These are also indicated in figure 2(b). Only a portion ( $y\in [-4,+4]$ , $z\in [0,{\rm\pi}]$ ) of the full computational domain ( $y\in [-6,+6]$ , $z\in [0,4{\rm\pi}]$ ) is shown.

In the earliest phase of the evolution (see figure 6 a, $t=7.5$ ) two vertical columns of counter-rotating vortices form in the ${\it\omega}_{x}$ -field. Each column is centred on the position of maximum growth rate $\pm y_{max}$ as discussed above and marked here by a thick grey vertical straight line. The columns or stacks of vortices result from the linear phase of the instability. The shape and distribution of the vortices are somewhat irregular due to the nature of the initial random perturbation that we have used to initiate the evolution. As an aid to the discussion, we also include vertical black dashed lines marking the limits of the linear instability region ( $y_{-},y_{+}$ ) and vertical solid black lines marking the limits of the predicted mixing range at $y=\pm y_{mix}$ (see figure 2 b).

As the vortices grow in strength, they begin to interact nonlinearly. They pair with neighbours, both above and below, forming dipolar ‘heads’ and ‘tails’. As these dipolar ‘heads’ and ‘tails’ develop, the vorticity in them tends to roll up. Consider figure 6(b) ( $t=10$ ). In the region of positive $y$ , what we are calling ‘dipolar heads’ have a positive vortex on the top and a negative vortex on the bottom. With this configuration, self-advection transports the dipolar head in the positive (northward) $y$ direction. Hence, these dipoles are on a trajectory to leave the region of linear instability. In the negative $y$ region the dipolar heads are those dipoles that form with negative vorticity on top and positive on the bottom. Hence, for them, the sense of self-advection is in the negative (southward) $y$ direction and thus they are also on a trajectory to exit the linear instability region. By $t=10$ , we see that some of the dipolar heads are crossing the boundaries of the linear instability region and one has already escaped. By this process, the perturbation is not confined to the linearly unstable region and mixing can occur well beyond that region. Also note that the vortices have formed dipolar tails and that these are oriented to propagate toward the equator at $y=0$ . Thus this mechanism of dipolar ‘head’ and ‘tail’ formation allows the instability to propagate beyond the linear instability region and to mix vigorously even within it.

By $t=20$ (figure 6 c), the strength of the vortical motions is decaying. The dipolar ‘heads’ have travelled to and beyond the predicted limits of the mixing range. As the dipoles approach these limits, they tend to be deflected as is already evident for one dipolar ‘head’ seen on the left in figure 6(b). In figure 6(c), we see that some of the dipolar ‘heads’ have left the predicted mixing region and have actually turned around and are oriented so as to return toward that region. There is also what we might call ‘debris’ from the destruction of the dipolar heads that have left, or tried to leave, the mixing region. By $t=50$ (figure 6 d), the ${\it\omega}_{x}$ field has decayed significantly from its peak strength, although there are some remaining regions of activity. Outside the mixing region in figure 6(d) there is some evidence of inertial wave radiation.

To conclude this section we note that the domain-integrated potential vorticity or absolute vorticity, i.e.  $\int \!Q(y)\text{d}y$ , is conserved. For the initial flow $Q=-\text{d}M/\text{d}y$ with $M(y)$ the momentum distribution, while for the predicted equilibrated flow the potential vorticity is $q=-\text{d}m/\text{d}y$ , with $m$ the momentum distribution which has $m=m_{c}$ between $y=-y_{mix}$ and $y=+y_{mix}$ . Hence

(3.11) $$\begin{eqnarray}\int _{-y_{mix}}^{+y_{mix}}Q(y)\text{d}y=-[M(y_{mix})-M(-y_{mix})]=-\int _{-y_{mix}}^{+y_{mix}}\frac{\text{d}m}{\text{d}y}\text{d}y=\int _{-y_{mix}}^{+y_{mix}}q(y)\text{d}y\end{eqnarray}$$

because in the prediction the level curve $m=m_{c}$ connects with $M(y)$ at $y=\pm y_{mix}$ (see figure 3 a). More generally, any $m(y)$ which connects with $M(y)$ at some $y=y_{l}$ and $y=y_{h}$ implies potential vorticity conservation but only certain choices imply momentum conservation.

4.1. Prediction?

For the uniform shear flow we face a problem: the initial $m$ -distribution $m=M(y)$ , as seen in figure 7, has a single maximum with no adjacent minima. If again we assume that the instability sets the $m$ -distribution to some constant $m=m_{c}$ over some $y$ -range, there is no obvious answer. Various equally plausible possible equilibria are sketched in figure 8. All satisfy the constraint of total absolute momentum conservation.

Figure 8.

Examples of possible momentum distributions of equilibrated flows for positive shear ${\it\Lambda}>0$ . In each panel the initial unstable $m$ -distribution is the parabola (thin solid line) with non-dimensional $m=M=y-y^{2}/2$ as in (4.2). Instability range is between non-dimensional $y=0$ and $y=1$ (see figure 7). (a) ‘Observed’ case (see text) with $m_{c}=0.232$ and $y_{l}=-0.464$ , $y_{h}=1.732$ (dashed), special case $m_{c}=3/8$ and $y_{l}=0$ , $y_{h}=3/2$ (thick solid) and a case with $m_{c}<0$ (dot-dashed). In each case (4.3) is satisfied. There is a positive jump at $y=y_{l}$ and continuity at $y=y_{h}$ with $m_{c}=M(y_{h})$ . The equality $2y_{h}+y_{l}=3$ is satisfied. (b) Two examples (thick dashed and thick solid) with $m=m_{c}$ between $y=y_{l}<0$ and $y=y_{h}>0$ and a positive jump at $y=y_{l}$ and negative jump at $y=y_{h}$ . (c) Continuous at $y=y_{l}\leqslant 0$ and negative jump at $y=y_{h}>0$ . Case $m_{c}=0$ , $y_{l}=0$ , $y_{h}=3$ is the thick dashed line. (d) Example (thick dashed line) with an internal jump at $y=0$ (equator) and two different $m_{c}$ -levels. Case $m_{c}=3/8$ and $y_{l}=0$ , $y_{h}=3/2$ (thick solid line) also shown in (a). For negative shear ${\it\Lambda}<0$ all these graphs have to flipped about the $y=0$ axis (see remarks in the caption of figure 7).

First note that always at least one discontinuity (a ‘jump’) in the equilibrated $m$ -profile would be required as in figure 8(a), where we show three possibilities that conserve $m$ . The jump is ‘positive’ ( from lower $m$ to higher $m$ ) on the left in the region $y\leqslant 0$ in each case at $y=y_{l}$ . A positive jump in the region $y>0$ would correspond to an unstable situation with ${\it\Phi}=-y\text{d}m/\text{d}y<0$ since $\text{d}m/\text{d}y=+{\it\delta}(y)$ . On the right, the adjusted profile connects at $y=y_{h}$ with the initial $m$ -profile with the condition that

(4.3) $$\begin{eqnarray}\int _{y_{l}}^{y_{h}}(M(y)-m_{c})\text{d}y=0,\end{eqnarray}$$

and in this case $m_{c}=M(y_{h})$ . We marked as ‘observed’ the case (dashed) with $m_{c}=0.232$ and $y_{l}=-0.464$ , $y_{h}=1.732$ which we will discuss further below. Also shown is the special case (solid) $m_{c}=3/8$ and $y_{l}=0$ , $y_{h}=3/2$ . Amongst the three possibilities shown in figure 8(a), this case has the smallest range over which $m=m_{c}$ with width $y_{h}-y_{l}=3/2$ .

In figure 8(b), we show two cases with a positive jump at $y=y_{l}<0$ and a negative jump at $y=y_{h}>0$ which also satisfy (4.3). In figure 8(c), we show the opposite of the scenario shown in figure 8(a), i.e. an adjusted profile which is continuous in $m$ at $y=y_{l}\leqslant 0$ and a negative jump at $y=y_{h}>0$ . The particular case with $m_{c}=0$ and $y_{l}=0$ , $y_{h}=3$ is the dashed line. Finally, in figure 8(d), we show yet another possibility with an ‘internal’ jump at $y=0$ but continuous at $y=y_{l}\leqslant 0$ and $y=y_{h}>0$ . The particular case with $m=m_{c}=-5/8$ between $y_{l}=-1/2$ and $y=0$ and $m=m_{c}=1/2$ between $y=0$ and $y_{h}=1$ is shown as the dashed line. This has the same ‘width’ ( $y_{h}-y_{l}=3/2$ ) as the other case shown in figure 8(d) (thick solid line with a jump at $y_{l}=0$ and continuous at $y_{h}=3/2$ ), which is also shown in figure 8(a).

All scenarios sketched in figure 8(a–c) imply conservation of total potential vorticity. Without going into details, it suffices to point out that the jumps can be represented using the Heaviside stepfunction multiplied by the appropriate jump amplitude. This means that in the $q$ -distribution there will be ${\it\delta}$ -functions at the jump locations. It is easily verified that this implies that $\int \!q(y)\text{d}y=\int \!Q(y)\text{d}y$ . Hence potential vorticity conservation does not provide an additional constraint with which a particular final equilibrium is singled out. In the same way it can also be shown that the dashed-line solution with the ‘internal’ jump in figure 8(d) also implies potential vorticity conservation. However, that solution can be ruled out because it has more energy than the initial state. In all cases, the final equilibrated state must have less energy than the initial state. Even taking into consideration this energy constraint, all examples shown in figure 8(a–c) are viable candidates. What nature will choose is thus not clear at all. The numerical simulations will provide the answer.

4.2. Numerical simulations

For the numerical simulations, it remains only to define the Reynolds number for the flow. Using the length scale $L={\it\Lambda}/{\it\beta}$ and time scale $T=1/{\it\Lambda}$ , the non-dimensional reciprocal of ${\it\nu}$ gives us the Reynolds number $\mathit{Re}={\it\Lambda}^{3}/({\it\beta}^{2}{\it\nu})$ for this flow.

The evolution of the profiles of vertically averaged absolute momentum $m$ and zonal velocity $u$ for the uniform shear flow is shown in figure 9. Once the instability begins, $m(y)$ goes through a rapid change, resulting by non-dimensional time $t=90$ in a relatively flat mixed region (figure 9 a). On the northern (positive $y$ ) end of this mixed region, the flat portion of $m(y)$ joins the initial condition without a jump. On the southern side, however, the mixed region joins the initial condition by a relatively steep jump at approximately $y=-1/2$ . As time proceeds, the mixed region tends to get smoother and flatter, while the slope of the jump tends to become shallower. This shallowing of the jump is a result of long-term viscous diffusion.

Figure 9.

Profiles of vertically averaged (a) absolute momentum $m$ and (b) zonal velocity $u$ for the uniform shear flow at the equator. The full computational domain for this simulation is $y\in [-4,+4]$ and $z\in [0,2{\rm\pi}]$ on a grid of $1537\times 1537$ points. Here we show only the $y\in [-1,+3]$ portion of the profiles. The Reynolds number of the flow is $\mathit{Re}={\it\Lambda}^{3}/{\it\beta}^{2}{\it\nu}=50$ k.

The evolution of $u(y)$ is illustrated in figure 9(b). As the $u(y)$ profile evolves, a step-like bump forms on the profile around non-dimensional $y\approx -1/2$ . The steepness of the positive (upward) slope of this bump increases rapidly between $t=30$ and $t=90$ . The steepness is greatest at $t=90$ , and then it slowly and continually becomes shallower from $t=90$ to the end of the simulation at $t=1000$ .

The effect of varying $\mathit{Re}$ on the equilibrium achieved after inertial instability is illustrated in figure 10. The curves drawn for various $\mathit{Re}$ in each panel of the figure correspond to non-dimensional time $t=200$ when the inertial instability is essentially equilibrated and slow viscous diffusion has not yet severely affected the profiles. Figure 10(a) demonstrates that as $\mathit{Re}$ increases, $m(y)$ becomes flatter and smoother in the ‘mixed region’, the region extends farther in the negative $y$ direction, and the steepness of the jump near $y\approx -1/2$ increases. The corresponding curves for $u(y)$ are shown in figure 10(b). The bump on the profile between $y\approx -1$ and $y\approx 0$ becomes increasingly step-like, with the jump becoming steeper with increasing $\mathit{Re}$ . The sharp gradient in $u$ at $y\approx -1/2$ corresponds to a large negative value of the (non-dimensional) potential vorticity $q=y-\text{d}u/\text{d}y$ . This was also observed in the numerical simulations of Griffiths who added uniform stratification to the problem (see Griffiths Reference Griffiths2003b ).

Figure 10.

Variation with $\mathit{Re}$ of profiles of vertically averaged (a) absolute momentum $m$ and (b) zonal velocity $u$ for the uniform shear flow at the equator. The curves all correspond to non-dimensional time $t=200$ by which the inertial instability has essentially equilibrated, but before viscous diffusion has had time to begin to smooth out the jump near $y\approx 0.5$ . The full computational domain is $y\in [-4,+4]$ and $z\in [0,2{\rm\pi}]$ on a grid of $2049\times 2049$ points. Only the $y\in [-1,+3]$ section of the computational domain is shown. The thick grey line in (a) is the prediction (4.4) with $m_{c}=0.232$ and $y_{l}=-0.464$ , $y_{h}=1.732$ , also marked as ‘observed’ in figure 8(a). The thin grey line in each plot corresponds to the initial condition. The Reynolds number is defined as $\mathit{Re}={\it\Lambda}^{3}/{\it\beta}^{2}{\it\nu}$ .

Based on these results and experience with equilibration on the $f$ -plane (see Kloosterziel et al. Reference Kloosterziel, Carnevale and Orlandi2007a ,Reference Kloosterziel, Orlandi and Carnevale b ; Carnevale et al. Reference Carnevale, Kloosterziel, Orlandi and van Sommeren2011, Reference Carnevale, Kloosterziel and Orlandi2013), we hypothesize that in the limit of infinite $\mathit{Re}$ the equilibrium profile of $m$ would be perfectly flat in the mixing region and there would be a jump discontinuity on the southern side of the mixing region at $y\approx -1/2$ . This would then correspond to one of the examples from figure 8(a). If $m_{c}$ is the level of $m$ in the mixing region and $y_{l}$ the location of the jump, then we hypothesis that the equilibrium profile would be

(4.4) $$\begin{eqnarray}\left.\begin{array}{@{}lll@{}}m=m_{c}, & u=m_{c}+y^{2}/2, & y\in [\,y_{l},y_{h}],\\ m=M(y)=y-y^{2}/2, & u=U(y)=y, & y\notin [\,y_{l},+y_{h}].\end{array}\right\}\end{eqnarray}$$

Assuming conservation of total absolute momentum according to (4.3), we find a cubic equation relating $y_{l}$ to $y_{h}$ for any given $m_{c}$ . Using the assumed continuity at $y=y_{h}$ so that $m_{c}=M(y_{h})$ this yields an exact relation: $2y_{h}+y_{l}=3$ . Based on the observed level $m_{c}\approx 0.232$ , from $M(y_{h})=m_{c}$ we obtain $y_{l}=-0.464$ and $y_{h}=1.732$ which agrees with the observed location of the jump near $y\approx -1/2$ . We have already sketched this solution in figure 8(a). It is drawn as the thick grey line in figure 10(a). This is not the solution of the smallest possible mixing range, nor does it correspond to any possible extremization of energy loss. There is no simple principle that we have been able to determine that selects this particular solution, even restricting consideration to the class of possibilities shown in figure 8(a).

The vortex dynamics behind the instability and its equilibration in the uniform shear flow are illustrated in figure 11. The evolution is similar to what we have seen in the Gaussian case but for the difference that in the ${\it\omega}_{x}$ field just one vertical column of vortices of alternating sign forms at the position of maximum growth as predicted by linear theory (figure 11 a). Each vortex pairs with its two neighbours to form dipolar heads/tails propagating in the negative/positive $y$ direction (figure 11 b). These dipolar structures propagate beyond the region of linear instability and fill out a region where the absolute momentum becomes well mixed (figure 11 c). As equilibration nears completion, there remain some relatively weak, long lived, vortex structures in the mixing region and some weak inertial waves beyond (figure 11 d). Clearly, the evolution seen in figure 11 is not symmetric as in the Gaussian jet case (see figure 6): we have only one region of instability from which intense activity emerges on the northern (positive $y$ ) side of the mixing region, away from the equator at $y=0$ , while far less is seen to the south of the equator where the jump develops.

Figure 11.

Contour plots of ${\it\omega}_{x}$ from a simulation of the uniform shear flow with $\mathit{Re}=50$ k at non-dimensional times (a)  $t=18$ , (b)  $t=22$ , (c)  $t=50$ (d)  $t=1000$ . The initial instability region, is bounded by the vertical dashed lines, at $y_{-}=0$ (the equator) and $y_{+}=1$ . The final limits of the inertial instability equilibration range, $y_{l}\approx -0.46$ and $y_{h}\approx +1.73$ , are indicated by the vertical solid black lines. Only a portion ( $y\in [-1.5,2.5]$ , $z\in [0,{\rm\pi}/2]$ ) of the full computational domain ( $y\in [-4,+4]$ , $z\in [0,2{\rm\pi}]$ ) is shown. The computation is done on a grid of $1537\times 1537$ points. Positive/negative contours are drawn as black/grey lines. The zero contour level is not drawn. The contour increments are (a) 0.42, (b) 1.5, (c) 3.1, (d) 0.15. In (a), the vertical solid grey line indicates the initial position of maximum growth rate $y_{max}=1/2$ according to linear theory. The Reynolds number of the flow is $\mathit{Re}={\it\Lambda}^{3}/{\it\beta}^{2}{\it\nu}=50$ k.

5.1. Eastward jet: small shift

In figure 12(a) we indicate what we mean by small: it is a combination of $y_{0}$ and $\mathit{Ro}$ such that the initial $M$ -distribution has $\text{d}M/\text{d}y>0$ in a range between the equator at $y=0$ and $y<y_{0}$ . This is also sketched in figure 13. The thick lines in both figures indicate this range. Locally the $M$ -distribution looks like a downward parabola, as for the uniform shear flow studied in § 4 (see figure 7). Hence the same ambiguity arises: how will the flow equilibrate? But since we now know how the linear shear flow adjusts, we can make a good guess. The answer is already sketched in figure 13: one can expect the development of a jump discontinuity south of the equator while momentum will be mixed to some constant level $m_{c}$ . A surprisingly good prediction for the mixing range and mixing level $m_{c}$ can be made using a linear shear matching procedure which is described in some detail in the appendix A.

An example is shown in figure 14. This is for the eastward jet with $\mathit{Ro}=4$ and a shift $y_{0}=1$ . In the appendix A we illustrate the procedure for the same jet but with a smaller shift ( $y_{0}=0.5$ , see figure 20). The idea is to approximate the initial momentum distribution $M$ around the position of maximal growth $y_{max}$ by

(5.5) $$\begin{eqnarray}M(y)\approx {\rm\Delta}M+Sy-By^{2}/2\equiv M_{S}.\end{eqnarray}$$

In figure 14(a) we show this approximation $M_{S}$ as a dotted line. It is obviously a downward parabola, just as the $M$ -profile of the uniform shear flow. At $y=y_{max}$ by construction we have $M_{S}=M$ and $M_{S}^{\prime }=M$ , $M_{S}^{\prime \prime }=M^{\prime \prime }$ ( $M^{\prime }\equiv \text{d}M/\text{d}y,M^{\prime \prime }\equiv \text{d}^{2}M/\text{d}y^{2}$ ). This means that also at $y_{max}$ the Rayleigh discriminant ${\it\Phi}_{S}\equiv -{\it\beta}y\text{d}M_{S}/\text{d}y={\it\Phi}$ . This is illustrated in figure 14(b). Even for this relatively large shift it is seen in the figure that ${\it\Phi}_{S}\approx {\it\Phi}$ over the entire instability range where ${\it\Phi}<0$ . In the appendix A it is explained why the flow that has initially a momentum distribution $M_{S}$ is expected to equilibrate with a mixing level $m_{c}={\rm\Delta}M+0.232(S^{2}/B)$ . The prediction only uses this predicted level and the assumption that just as for the linear shear flow a jump occurs south of the equator ( $y=y_{l}<0$ ), as in figure 8(a) (‘observed’) or for example figure 10(a), while at $y=y_{h}>0$ on the right-hand side of the mixing range $m_{c}=M$ . The location of the jump is then determined by momentum conservation, i.e. (4.3) is satisfied. The prediction is drawn as a thick grey curve in figure 14(c). It is seen that around $t=300$ when the flow has equilibrated, the vertically averaged momentum indeed matches the prediction remarkably well except near the jump position. In the simulation we used $\mathit{Re}=400$ k. Much higher $\mathit{Re}$ are expected to reveal a ‘sharpening’ of the jump, just as is seen in figure 10(a).

Figure 14.

(a) Initial momentum $M$ (solid grey line) and the approximation $M_{S}$ (dotted line) for the eastward flowing jet with $\mathit{Ro}=4$ and shift $y_{0}=1$ . The approximation $M_{S}$ is given by (5.5) with non-dimensionalized ${\rm\Delta}M=1.11$ , $S=4.91$ , $B=4.58$ (see (A 5) in the appendix A). The location of fastest growth $y=y_{max}=0.53$ is indicated by a ●. The thick black part of $M$ indicates the instability region $y\in [0,0.88]$ . (b) Non-dimensionalized Rayleigh discriminant ${\it\Phi}$ (solid line) and the approximation ${\it\Phi}_{S}=-{\it\beta}y\text{d}M/\text{d}y$ (dotted line). The location of fastest growth $y=y_{max}=0.53$ is indicated by a ●. In (c) and (d) the prediction based on the linear shear dynamics (see appendix A) for the equilibrium $m$ and $u$ -field is shown as a thick grey curve. The black curve in (c) and (d) is the result at non-dimensional time $t=300$ of a numerical simulation of the equilibration of the eastward Gaussian jet with (c) absolute momentum $m$ vertically averaged and (d) velocity $u$ vertically averaged. At time $t=300$ the inertial instability has essentially equilibrated. The full computational domain is $y\in [-4,4]$ by $z\in [0,4{\rm\pi}]$ with 1025 grid points in each direction. Only a section of the $y$ -range is shown. The Reynolds number for the flow is $\mathit{Re}=|U_{0}|L/{\it\nu}=400$ k. In (d) the $u$ -field has been normalized with $\mathit{Ro}$ , i.e. we show $u/|\mathit{Ro}|$ so that the peak velocity at $t=0$ is $U(y_{0})/|\mathit{Ro}|=1$ .

The jump discontinuity that is seen to have formed in figure 14(c) was schematically indicated in the upper left part of figure 13. If we compare this with the results for the linear shear flow shown in figure 10(a), there is clearly little qualitative difference. But the adjusted velocity field seen in figure 14(d) for the Gaussian jet is obviously very different from the adjusted linear shear flow seen in figure 10(b). This indicates that for the adjustment of the inertial instability the finer details of the flow away from the initial instability region do not matter much. For example, in figure 14(d) the mixing has produced a profile which connects to the Gaussian profile to the right of the peak velocity ( $y=y_{0}$ ) where the flow is decreasing with distance. For the uniform shear flow in figure 10(b) the mixing has proceeded to the right to connect with the linearly increasing flow. But in both cases the initial momentum distributions and the Rayleigh discriminant in the region of instability and ultimate mixing region ‘look’ the same and this is all that matters.

5.2. Eastward jet: critical shift and large shift

From the diagram in figure 13 it should be clear that larger $\mathit{Ro}$ imply a larger ‘bump’ added to the downward parabola $-{\it\beta}y^{2}/2$ , the sum of which is the absolute momentum. It should also be clear that if the eastward jet with some fixed large $\mathit{Ro}$ is shifted to larger $y_{0}$ than just discussed, the mixing hypothesis (constant $m_{c}$ over some range and conservation of total absolute momentum) implies that for some $y_{0}$ the mixing range extends exactly to the equator but not beyond and no jumps south of the equator are expected. We call this the ‘critical shift’ although it is really a specific combination of $y_{0}$ and $\mathit{Ro}$ . Figure 13 shows an example. Experimentation with the mixing recipe reveals that the smallest shift for which the critical case occurs is $y_{0}\approx 2.2$ with $\mathit{Ro}\approx 1.98$ . For $y_{0}$ increasing monotonically above the value 2.2, the corresponding critical $\mathit{Ro}$ increases monotonically as well. For example, when $y_{0}=5$ the critical $\mathit{Ro}$ is $\mathit{Ro}=20.6$ . The results of numerical simulation for this particular critical combination of $y_{0}$ and $\mathit{Ro}$ are shown in figure 15(a,b). Figure 15(a) shows that the inertial instability has mixed the momentum between the equator and a position to the right of the peak of the jet, as seen in figure 15(b). The agreement with our prediction (thick grey lines) is excellent. This example is representative for all possible critical cases with $y_{0}>2.2$ .

Figure 15.

Plots of the vertically averaged absolute momentum $m(y)$ and $u(y)$ showing the initial (thin grey curve) conditions and the state after a time sufficient to allow nearly complete absolute momentum mixing (solid black curve), for the eastward flowing Gaussian initial jet on the ${\it\beta}$ -plane with: (a) and (b) a ‘critical shift’ example (see text) with $y_{0}=5$ and $\mathit{Ro}=20.6$ , (c) and (d) a ‘large shift’ example with $y_{0}=10$ and $\mathit{Ro}=40$ . In each panel the thick grey curve is the prediction based on our momentum mixing scheme with no discontinuities. For the critical case the mixed level in (a) is $m_{c}\approx 0$ . At time $t=90$ (a,b) and $t=70$ (c,d) the inertial instability has equilibrated. The full computational domain for (a) and (b) is $y\in [-3,9]$ by $z\in [0,4{\rm\pi}]$ with 1025 grid points in each direction. For (c) and (d) the $y$ -range is $y\in [2,14]$ . In all panels only a section of the $y$ -range is shown. The Reynolds number is $\mathit{Re}=50$ k. In (b) and (d) we show $u/|\mathit{Ro}|$ so that the peak velocity at $t=0$ is $U(y_{0})/|\mathit{Ro}|=1$ .

What we call ‘large’ next is somewhat arbitrary, but simply put: it is a combination of $y_{0}$ and $\mathit{Ro}$ so that the mixing does not reach the equator and no jumps are expected. One can see from figure 15(a) that if $\mathit{Ro}$ is slightly lowered from $\mathit{Ro}=20.6$ , the peak amplitude around $y_{0}=5$ in the initial momentum distribution (thin grey line) becomes a bit lower while barely affecting the depth of the ‘valley’ around $y=3$ because of the rapid fall-off of the Gaussian jet amplitude with distance from $y_{0}$ . Momentum can then be conserved by mixing between some $y=y_{l}>0$ and a position $y_{h}>y_{0}$ with an $m_{c}$ lower than in figure 15(a) so that (2.7) is satisfied. Instead of showing that, we show in figure 15(c,d) a case with $y_{0}=10$ and large $\mathit{Ro}=40$ . It is seen again that the agreement between the numerical results and the prediction is very good. This case was also sketched in figure 13. One may find this an extremely high Rossby number, but it is really a matter of definitions. Dimensionally $y_{0}=10$ corresponds to a distance $10L$ with $L$ the width of the Gaussian jet. The Coriolis parameter at $y_{0}$ is $f_{0}={\it\beta}y_{0}$ . If instead of $\mathit{Ro}=U_{0}/{\it\beta}L^{2}$ we choose $\mathit{Ro}=\mathit{Ro}_{f_{0}}=U_{0}/f_{0}L$ then with the shift $y_{0}=10L$ we would have had $\mathit{Ro}_{f_{0}}=4$ .

The reason for showing the example seen in figure 15(c,d) and not a more ‘reasonable’ one, is twofold. First, because although all in this study is formulated in the context of equatorial ${\it\beta}$ -plane dynamics, things quickly start to look like mid-latitude $f$ -plane dynamics. Figure 1(b) for example used the absolute momentum on the $f$ -plane, i.e.  $M=U(y)-f_{0}y$ (with $U$ also a Gaussian jet) but it is not unlike figure 15(c). Secondly, if figure 15(a) is compared with 15(c), one sees very different initial momentum distributions. But the mixing through inertial instability leads to adjusted, inertially stable flows that differ very little, as comparison of panel (b) with panel (d) of figure 15 reveals. Note that although ‘ $f$ -plane-like’ adjustments are observed, the length scale $L$ can be anything, say a mere 10 km. Then a shift of 100 km (non-dimensional $y_{0}=10$ ) would hardly be cause to start considering mid-latitude $f$ -plane dynamics, but with such small $L$ the Rossby number $\mathit{Ro}=U_{0}/{\it\beta}L^{2}$ can be quite large, even for small $U_{0}$ . Therefore the adjustment near the equator can be quite similar to that found at high latitudes.

5.3. Westward jet: small and large shift

As for the linear shear flow discussed in § 4, the adjustments for the eastward jet are triggered from a single region of instability. But in § 3 we saw that the westward Gaussian jet centred on the equator has two regions of instability where the Rayleigh discriminant ${\it\Phi}<0$ (see figure 2 b). They meet at the equator and because in both regions the rate of normal-modes growth is equal and the initial absolute momentum is symmetric about the equator (see figure 3 a), the prediction was easy (thick line in figure 3 a). The secondary motions associated with the instability seen in figure 6(a) amplify at equal rates and propagate from both regions towards the equator and away from the equator, mixing the momentum and a new, stable flow emerged. The prediction was well-verified, e.g. see figure 5.

If the symmetry is broken by shifting the jet away from the equator, we still have two regions of instability provided $\mathit{Ro}<-1/2$ . But they become separated as sketched in figure 12(b). In figure 17(a) we show the Rayleigh discriminant for the westward jet with $\mathit{Ro}=-4$ and the northward shifts $y_{0}=0.4,0.8$ and $1.6$ . Clearly seen is that the regions of instability, where ${\it\Phi}<0$ , become ever more separated. To the south ( $y<0$ ) the most negative value of ${\it\Phi}$ decreases in amplitude with increasing shift, to the north ( $y>0$ ) it first increases and then decreases. As mentioned in § 2 normal-modes growth $\exp ({\it\gamma}t)$ of the inertial instability is expected to have ${\it\gamma}<\sqrt{\max _{y}(-{\it\Phi})}$ with the maximal growth rate ${\it\gamma}_{max}=\sqrt{\max _{y}(-{\it\Phi})}$ attained in the limit of infinite Reynolds number. Therefore the ratio of the theoretical upper limits on growth, that is ${\it\gamma}_{max}(y>0)/{\it\gamma}_{max}(y<0)$ for the shifted Gaussian jet, tells us how disparate the rates of growth are. This is shown in figure 17(b). Further note that figure 17(a) shows that, although the regions of instability become more separated, with increasing shift the position of fastest growth on the southern side ( $y=-y_{max}$ in figure 6 a) moves towards the equator and the position of fastest growth on the northern side ( $y=+y_{max}$ in figure 6 a) moves away from the equator. Also on both sides the widths of the instability regions decrease.

Figure 16.

Diagram showing a ‘side view’ of the absolute momentum distribution $M=U(y)-{\it\beta}y^{2}/2$ for the shifted westward jet. Two possibilities are shown: a ‘small shift’ and a ‘large shift’ where equilibration is reached through mixing to some level $m_{c}$ as sketched in figure 1(b) for the $f$ -plane and in figure 13 for the shifted eastward jet. For small shifts and sufficiently large $|\mathit{Ro}|$ two regions of instability with different rates of normal-modes growth exist initially, separated by an inertially stable region north of the equator at $y=0$ , as sketched in figure 12(b). For very small shifts no prediction appears possible (see figure 18 a,b) while for modestly small shifts a prediction is possible (see figure 19).

Figure 17.

(a) The Rayleigh discriminant ${\it\Phi}$ for the westward Gaussian jet with $\mathit{Ro}=-4$ and northward shifts $y_{0}=0.4,0.8$ and 1.6. (b) Ratio of maximal normal-modes growth rate (at the positions marked by ●), i.e. ${\it\gamma}_{max}(y>0)/{\it\gamma}_{max}(y<0)$ . For larger shifts $y_{0}\gtrapprox 2$ the southern instability region disappears and $f$ -plane-like adjustment occurs, marked by ‘large shift’ in figure 16.

In figure 18 we show the equilibration of the westward jet for the shifts $y_{0}=0.4$ and 0.8 and fixed Rossby number $\mathit{Ro}=-4$ . In each case the Reynolds number was  $\mathit{Re}=50$ k. In figure 18(a) the initial growth rate of the instability is in the northern region about 1.3 times faster than in the southern region. At $t=50$ to the north a tendency to develop a constant mixing level is observed while some significant changes are also seen in the southern part. However at the later times $t=150$ and $t=350$ to the south an almost flat $m$ -distribution is seen to have formed. This seems to indicate that there the flow tends towards having a constant $m=m_{c}$ but it smoothly transitions to the northern part where no flat section has formed after all. This picture did not simplify on increasing the resolution of the simulation from $513\times 513$ gridpoints to $1025\times 1025$ , and the vortices involved in mixing do seem to be resolved here. We conclude that in this example the growth rates and the regions of instability are too close to make a simple prediction possible.

Figure 18.

Plots showing the evolution of vertically averaged absolute momentum $m(y)$ for the westward flowing Gaussian jet with $\mathit{Ro}=-4$ and (a)  $y_{0}=0.4$ and (b)  $y_{0}=0.8$ with $\mathit{Re}=50$ k. The full computational domain is $z\in [0,1.5{\rm\pi}]$ and $y\in [-6,+6]$ with 1025 grid points in each direction. Only a portion of the $y$ -domain is shown. In both panels the vertical grey line indicates the position $y_{0}$ of the jet, the equator is at $y=0$ . In (a) with $y_{0}=0.4$ the initial rate of normal-modes growth north of the equator is 1.36 faster than that in the region south of the equator, in (b) with $y_{0}=0.8$ the ratio is 1.96. The thick grey line in (b) is the prediction based on a combination of linear shear matching in the southern region with a jump north of the equator and in the northern region momentum mixing to a constant level $m_{c}$ between $y=y_{l}>0$ and $y=y_{h}$ (see appendix A). In this example the predicted jump position coincides with $y=y_{l}$ . For larger shifts $y_{0}$ the two regions separate and the prediction becomes very accurate at high resolution (see figure 19).

For the larger shift $y_{0}=0.8$ in figure 18(b) we see that at $t=50$ (dotted line) to the north the flow is close to having a mixed region with a constant $m=m_{c}$ . Locally it is the equilibrium predicted in figure 15(c) for the eastward jet far from the equator according to (2.7). But at $t=50$ no changes south of the equator are visible in figure 18(b). Only much later (at $t=400$ ) the entire flow is equilibrated and a major change is seen in the southern region but by then the flow to the north, which was at $t=50$ mixed to a constant $m_{c}$ , has also been changed considerably.

The thick grey line in the figure 18(b) marked with ‘prediction?’ is described in the appendix A. It is a combination of two predictions, namely: (a) the northern instability region equilibrates through momentum mixing as sketched in figure 1(b) or as in figure 15(c); and (b) the southern instability region equilibrates through a jump development north of the equator, similar to a mirror image of what we showed in figure 14(c) for the eastward jet. Treated as two separate problems, assuming that the other region of instability remains unchanged, the prediction becomes very accurate when the two mixing ranges do not overlap. Figure 18(b) is an example when the two hypothetical ranges touch at the mixing boundary that they share. In this case, the prediction is not good. But, for slightly larger shifts, the two mixing regions separate and then the prediction rapidly becomes quite accurate. An example is shown in figure 19(a) where we used $y_{0}=1.2$ . At $t=50$ (dotted line) the northern region has equilibrated with constant $m=m_{c}$ with no changes south of the equator, while at the much later time $t=700$ the southern instability shows signs of having caused a jump north of the equator. But no flat mixed region is seen otherwise. This is, however, due to the fact that the number of grid points used here ( $2049\times 2049$ ), although sufficient to resolve the vortices over most of the domain, did not completely resolve those in the small southern instability region. However, since the two mixed regions do not overlap, they can be treated separately numerically.

Figure 19.

Plots showing the evolution of vertically averaged absolute momentum $m(y)$ for the westward flowing Gaussian jet with $\mathit{Ro}=-4$ and $y_{0}=1.2$ . In this example the growth rate of the instability on the northern side is 3.15 times faster than that on the southern side (see figure 17). (a) At $t=50$ the northern region of instability has mixed to a constant level $m=m_{c}$ while in the southern instability region no changes are visible. At $t=700$ the instability to the south is near equilibrium but the mixing has not produced a constant mixing level $m=m_{c}$ . Since the growth is much slower in this region the equilibrium already reached at $t=50$ to the north has diffused away from the horizontal level at the later time $t=700$ . (b) The equilibrium at $t=700$ in the southern region as a function of $\mathit{Re}$ . The computational domain was $z\in [0,{\rm\pi}/2]$ and $y\in [-1.6,+1.6]$ with 2049 grid points in each direction. With this much higher resolution the upward slope seen in (a) has disappeared. The prediction is described in the appendix A. The thin grey line in each plot corresponds to the initial condition (IC).

In figure 19(b) we show the adjustment in the southern part using a much reduced domain and increased resolution, concentrated about the southern instability region of figure 19(a) (details are in the caption). It is seen that for $\mathit{Re}=50$ k the slope in the $m$ -profile seen in figure 19(a) at $t=700$ has disappeared in figure 19(b). Further increases of $\mathit{Re}$ increases the sharpness of the downward jump but for all $\mathit{Re}$ clearly the mixed level $m_{c}$ is very close to the predicted level (thick grey line). Thus, we conclude that if the shift is sufficiently large, generally the two regions of instability equilibrate at different rates in different fashions: a rapid adjustment in the northern instability region with a mixed level $m_{c}$ and mixing range $y\in [\,y_{l},y_{r}]$ determined by (2.7), as for the eastward jet shown for example in figure 15(c), and an adjustment with a downward jump north of the equator following the prediction from the linear shear matching procedure described in the appendix A.