8.1 Convective penetration

Due to its wide applicability in astronomy and geophysics, there is a large body of literature on convective penetration. Much of this work, particularly in a solar context, is concerned with the structure of the overshoot region and how the penetration depth varies with the vigor of the convection and the stiffness of the transition from subadiabatic to superadiabatic stratification. Figure 22

illustrates the structure of the overshoot region at the base of the solar envelope as suggested by Zahn (1991

). In the convection zone, the radial entropy gradient, $-- dS/dr$ , is negative but nearly adiabatic due to the efficient mixing of entropy by turbulent convection. The convective enthalpy flux is positive (outward) and the radiative heat flux normalized by the total flux, $Lo . /4pr2$ is less than unity. To a good approximation, the normalized convective enthalpy flux and radiative heat flux sum to unity, with smaller contributions from the other terms in Equation (3

In many theoretical studies, the base of the convection zone is defined as the point where $-- dS/dr$ changes sign and becomes positive (subadiabatic). The inertia of convective downflows takes them beyond this point into the stably-stratified interior. Here the enthalpy flux becomes negative (inward) and the outward radiative flux must increase to compensate. Downward motions will be quickly decelerated by buoyancy but the turbulent mixing may still be efficient enough to establish a nearly adiabatic penetration region where $-- dS/dr > 0$ . Eventually, downflows will be decelerated enough such that their effective Péclet number, $Pe = U L/k$ , becomes small and turbulent mixing becomes inefficient relative to thermal diffusion. This occurs in a thin thermal adjustment layer where the enthalpy flux falls to zero and the stratification becomes strongly subadiabatic. Deeper in the interior, the radiative heat flux carries the entire solar luminosity.

Figure 22:

A schematic diagram illustrating the radial entropy gradient, $-- dS/dr$ , the convective enthalpy flux, $EN F$ , and the radiative heat flux $RD F$ near the base of the convection zone (see Equation (3

) and Appendix A.3). Each quantity is plotted on a horizontal axis (increasing toward the right) as a function of radius (vertical axis). The radiative flux is normalized with respect to the total solar flux, $L /4pr2 o .$ . Four regimes are indicated as discussed in the text (after Zahn, 1991

Sometimes a distinction is made between convective overshoot and convective penetration. The former is used to describe any convective motions which are carried into a region of stable stratification by their own inertia. By contrast, the latter term often has a more specific meaning, implying that the convection is efficient enough to establish a nearly adiabatic penetration region as indicated in Figure 22

. From the perspective of solar structure modeling and helioseismic probing, it is often more convenient to define the base of the convection zone as the bottom of the well-mixed, nearly adiabatic penetration region rather than where the entropy gradient changes sign.

The presence of a nearly adiabatic penetration region in the Sun is currently a matter of some debate. Although many early models and relatively low-resolution 2D and 3D simulations produced a true penetration region where $-- d S/dr > 0$ (reviewed by Brummell et al., 2002b ; Rempel, 2004 ), recent high-resolution simulations of turbulent penetrative convection by Brummell et al. (2002b ) exhibited only strongly subadiabatic overshoot. They attributed the absence of a nearly adiabatic penetration region to the small filling factor of downflow plumes, which dominate the flow field in turbulent parameter regimes (see Section 5.2). However, reduced models based on the dynamics of intermittent plumes suggest that such numerical simulations may exhibit more adiabatic penetration if they could achieve more solar-like parameter regimes (Zahn, 1991 ; Rempel, 2004 ). In particular, higher Péclet numbers and a lower imposed heat flux may modify the balance between advective and diffusive heat transport enough to produce a nearly adiabatic stratification.

Another challenge to numerical simulations of penetrative convection is achieving a high stiffness parameter $St$ , which is a measure of the subadiabatic stratification in the stable zone relative to the superadiabatic stratification in the convection zone. In the Sun this ratio is roughly $105$ whereas simulations consider values of at most $10 -100$ . Thus, the depth of penetration, $Dp$ , in simulations is artificially high and much work has focused on establishing scaling relations between $Dp$ and $S$ in order to extrapolate the results to solar conditions. Analytic estimates by Hurlburt et al. (1994 ) suggest that the extent of the nearly adiabatic penetration region, if present, scales as $S-1 t$ whereas the depth of the thermal adjustment layer scales as $- 1/4 St$ . Numerical simulations are generally consistent with these scaling estimates (Hurlburt et al., 1994 ; Singh et al., 1995 ; Brummell et al., 2002b ). However, Rogers and Glatzmaier (2005a ) have recently achieved stiffness values of over 500 in high-resolution simulations of 2D penetrative convection and they find a much shallower scaling law, $D ~ S -0.04 p t$ for $S > 10 t$ . When extrapolated to solar conditions, most simulations and models imply penetration depths ranging from about $0.01 -1$ pressure scale heights $HP$ , implying a $Dp$ of a few percent of the solar radius or less (see, e.g., Rempel, 2004 ; Stix, 2002 ). By comparison, upper limits from helioseismology suggest that the overshoot region is no more than about $0.05HP$ , which is less than $0.01Ro .$ (Section 3.6). Helioseismic inversions can also set limits on how abruptly the entropy gradient changes at the base of the convection zone, ruling out a very thin thermal adjustment layer (Monteiro et al., 1994 ; Basu et al., 1994 ; Roxburgh and Vorontsov, 1994 ).

Brummell et al. (2002b ) also considered the variation of the penetration depth with rotation and latitude, under the f-plane approximation. They found that rotation generally has a stabilizing effect because plumes are tilted away from the vertical by turbulent alignment and weakened by vortex interactions. Similar results were also reported by Julien et al. (1996a , 1999 ); see Section 5.2. The penetration depth was greatest at the equator and poles, and least at mid-latitudes. The smaller penetration at mid-latitudes relative to high latitudes was attributed to turbulent alignment because tilted plumes have less downward momentum. The enhanced penetration at low latitudes was attributed to the formation of horizontal convective rolls which are analogous to the north-south aligned downflow lanes typically seen in global convection simulations (Section 6.2). Global simulations of penetrative convection by Miesch et al. (2000 ) do indeed exhibit deeper penetration at the equator, but there is less evidence for enhanced penetration at the poles in turbulent parameter regimes. However, the simulations by Miesch et al. (2000 ) used a realistic value for the solar luminosity so it was impractical to cover a full thermal equilibration timescale ( $~ 105 yr$ ; see Section 5.1). Thus, any conclusions made about the detailed structure of the overshoot region must be regarded as tentative.

Investigating convective penetration with global models remains an important challenge for the near future. Although global models can say little about the thermal structure of the overshoot region at present, they have already produced provocative and robust results regarding its dynamics. In particular, they have indicated that penetrative convection in the Sun is likely to induce equatorward meridional circulation and poleward angular momentum transport in the overshoot region (see Sections 6.3 and 6.4).

Another aspect of penetrative convection which has important implications for solar interior dynamics is the generation of gravity waves. Figure 23 illustrates wave excitation in simulations of penetrative convection by Rogers and Glatzmaier (2005b ). The geometry is a 2D circular annulus with the inner boundary placed very near the origin to minimize spurious wave reflection. Gravity waves appear as rings of vorticity in the stable zone propagating outward. This outward phase velocity implies an inward group velocity, and is therefore consistent with wave generation at the base of the convection zone (see Appendix A.7).

Figure 23:

Movie. The vorticity field is shown in a simulation of penetrative convection in a circular annulus (from Rogers and Glatzmaier, 2005b

) (courtesy T. Rogers).

Although gravity waves are present in all simulations of penetrative convection, little is known about the details of wave excitation in the Sun. Unless steps are taken to avoid it, numerical simulations generally suffer from wave reflection at the lower boundary and imposed horizontal periodicities which can substantially alter the spectra, energetics, and transport properties of the waves. Furthermore, obtaining a reliable estimate of gravity wave amplitudes and spectra in a high-resolution simulation of penetrative convection is not a trivial undertaking (e.g., Dintrans and Brandenburg, 2004 ). The most straightforward method is based on spectral transforms of the velocity or density field in space and time, but this can be unwieldy in a 3D simulation because it requires storing a substantial volume of data at a high temporal cadence and a long enough duration to achieve stable statistics. To date, most investigations of gravity wave excitation in simulations of penetrative convection have been restricted to 2D flows (Hurlburt et al., 1986 ; Andersen, 1996 ; Dintrans et al., 2003 ; Kiraga et al., 2003 ; Rogers and Glatzmaier, 2005a ,b ). Theoretical estimates of wave excitation are sensitive to assumptions made about the structure of the convection which are difficult to justify (Goldreich and Kumar, 1990 ; Fritts et al., 1998

; Kumar et al., 1999

Despite this uncertainty, some general comments can be made. We expect that the gravity wave spectra will peak at spatial and temporal frequencies which correspond to the characteristic scales of the convection which drives them. These are currently uncertain but may be estimated from numerical simulations (Section 6.2). Modes with very small wavelengths ( $< 1 Mm$ ) will be efficiently dissipated by thermal diffusion while modes with horizontal phase velocities comparable to the local differential rotation will be filtered out by critical level absorption and radiative diffusion (Fritts et al., 1998 ; Kumar et al., 1999 ; Talon et al., 2002 ). If the motions are indeed gravity waves, their frequencies will be bounded from above by the Brunt-Väisälä frequency, $N$ , which corresponds to a period of a few hours in the solar interior. However, since the Sun is rotating and magnetized, we might expect a wide variety of waves to be generated by penetrative convection, including inertial gravity waves, Rossby waves, and Alfvén waves. Characteristic velocity amplitudes will vary substantially with radius but may be $~ 1- 10 m s-1$ near the overshoot region based on estimates for the vertical velocity in downward plumes, which may reach $-1 100 m s$ , and a moderate conversion efficiency.

No discussion of penetrative convection would be complete without some mention of transport processes. It is well established that turbulent penetrative convection can efficiently pump magnetic fields out of the convection zone into to the overshoot region, and possibly deeper (Brandenburg et al., 1996 ; Tobias et al., 1998 , 2001 ; Dorch and Nordlund, 2001 ; Ziegler and Rüdiger, 2003 ). This is thought to play an integral role in the solar dynamo by continually supplying the tachocline with disordered field which can then be organized and amplified by rotational shear (Section 4.5). Transport of chemical tracers by penetrative convection and the waves it generates can has important implications for solar structure models and spectroscopic measurements of stellar compositional abundances (Montalbán, 1994 ; Schatzman, 1996 ; Hurlburt et al., 1994 ; Pinsonneault, 1997 ; Fritts et al., 1998 ; Brummell et al., 2002b ; Ziegler and Rüdiger, 2003 ). Furthermore, angular momentum transport by gravity waves has important implications for understanding the structure and evolution of the solar internal rotation profile as we will discuss further in Sections 8.4 and 8.5.

We emphasize that convective penetration in the Sun is a very intermittent process, dominated by extreme, impulsive events; particularly strong plumes or ensembles of plumes which penetrate deeper than average and then quickly lose coherence. A jackhammer is a better analogy than a drill. Thus, the transport of magnetic fields, chemical tracers, and momentum, is generally deeper than might be expected from average measures such as the mean stratification or the mean kinetic energy density (e.g., Brummell et al., 2002b ).