Dynamics of nonlinear Alfvén waves in the shallow water magnetohydrodynamic equations

The large-scale hydrodynamics of the solar tachocline have been modeled by a version of the well-known shallow water equations augmented to include the effects of magnetism. Due to the conservation of magnetic flux, these shallow water magnetohydrodynamic equations have an additional constraint that the classical shallow water equations (without magnetism) do not. This restricts the set of possible initial conditions, but in itself does not provide information about the dynamics. The linear theory of wave motion on both the f plane and $\beta$ plane is relatively well documented. We thus seek to explore the dynamics of nonlinear waves on the magnetohydrodynamic f plane. By initializing with a circular vortex and no free surface displacement, we make contact with a substantial literature in geophysical fluid dynamics, satisfy the magnetohydrodynamic constraint, and provide a means to generate all possible wave species through the adjustment process. We find that for Cowling numbers typical of the solar tachocline, the response is dominated by zonally propagating Alfvén waves, which take the form of vortices. These vortices are expressed in the velocity, free surface, and magnetic field variables. We explore how nonlinearity and variation in the Cowling number affect these dynamical structures, finding that nonlinearity warps the axially symmetric Alfvén wave modes of linear theory. Moreover, while the waves maintain a fixed shape during pure propagation, we find that wave shape is significantly modified during collisions. The extent of warping during collisions is amplitude dependent, and hence a nonlinear effect, albeit one that is fundamentally different from soliton models of nonlinear waves in shallow water. We explore parameter ranges outside those relevant to our Sun's tachocline and find that our results are robust over significant portions of parameter space. In the limit of rapidly rotating stars (i.e., rates 100 times larger than those associated with our Sun) we find that the Alfvén waves destabilize over time.

Figure 1

Evolution of the kinetic energy in the baseline case (BASE). Panels (a)–(d) show times $t=1.0,1.67,2.34,3.01$, respectively.

Figure 2

Plots of (a) the kinetic energy, (b) the enstrophy, and (d) the PMFE in the baseline case (BASE). (c) The kinetic energy with isolines of the PMFE overlain. The time shown, $t=3.01$, corresponds to Fig. 1.

Figure 3

Space-time plot of the kinetic energy profile along the $y=0$ transect showing that during collisions wave shape is altered, but during propagation between collisions the wave shape remains unchanged.

Figure 4

Comparison of the kinetic energy in the baseline case (BASE) (a) before and after (b) one, (c) two, and (d) three head-on collisions. Panels (a)–(d) show times $t=6.02,10.2,21.4,25.6$, respectively.

Figure 5

Comparison of the kinetic energy distributions in the propagating waveforms (a) in the baseline (BASE) case and (b) when the polarity of the initial vortex is reversed (NEGATIVE $U_0$) at $t=3.01$.

Figure 6

Comparison of the kinetic energy distributions in the propagating waveform (a)–(c) before and (d)–(f) after collision as amplitude changes. Initial velocity reduced by a factor of 10 (case SMALLEST $U_0$) shown on left (a), (d); the baseline (BASE) case shown in the middle (b), (e); and the initial velocity doubled case (LARGE $U_0$) shown on the right (c), (f). Note the nearly perfect circular symmetry in the SMALLEST $U_0$ case, in panel (a), which is only slightly disturbed after the collision, in panel (d). The top row (a)–(c) shows rightward propagating waveforms before a collision ($t=3.01$), while the bottom (d)–(f) shows the waveform after one collision ($t=10$).

Figure 7

Comparison of the kinetic energy distributions in the propagating waveform in (a) the baseline (BASE) case and (b), (c) with the value of $L_0$ halved (BSM). Panel (c) shows panel (b) rescaled by the same factor as $L_0$ thereby demonstrating that in the magnetically dominated regime the dynamics are essentially unaffected by the vortex length scale. Times shown are $t=3.01$ for panel (a) and $t=6.01$ for panels (b) and (c).

Figure 8

Plots of the kinetic energy evolution over the course of the simulation when the magnetic field strength is a tenth of that in the baseline case (LOW $B_{R_0}$). A blue grid is superimposed on the image to show the outward migration of the spiral arms of high KE. Panels (a)–(f) show times $t=0,0.04,0.05,0.06,0.07,0.08$, respectively.

Figure 9

Shaded contours of the kinetic energy with three contours of PMFE superimposed. The magnetic field strength is a tenth of that in the baseline case (case LOW $B_{R_0}$). Panels (a)–(d) show times $t=0.04,0.053,0.067,0.8$, respectively.

Figure 10

Plots of the kinetic energy evolution over the course of the simulation when the magnetic field strength is a tenth and $U_0$ is a half of those in the baseline case (LOW $B_{R_0}$ LOW $U_0$). Panels (a)–(f) show times $t=0.03,0.1,0.17,0.23,0.3,0.37$, respectively.

Figure 11

Evolution of the kinetic energy in the case with $f$ increased 100-fold (HIGH $f$) at $t=1.33,1.67,2,2.33,2.67,3$ in panels (a)–(f) respectively. Three PMFE contours are superimposed in blue. Note the variability of the waveform shape in this case. Also note that no collisions occur within the time frame shown.

Figure 12

Space-time plot of the kinetic energy profile along the $y=0$ transect showing the breakdown of the Alfvén waves for the case with $f$ increased 100-fold (HIGH $f$).

Figure 13

Evolution of the kinetic energy in the case with $f$ increased 100-fold (HIGH $f$) (a) before collision, (b) after one collision, and (c) after two collisions. Three PMFE contours are superimposed in blue. Panels (a)–(c) show times $t=4.7,12.6,20.4$, respectively.