Nonlinear Vertical Scale Selection in Equatorial Inertial Instability

Transcription

1 1APRIL 003 GRIFFITHS 977 Nonlinear Vertical Scale Selection in Equatorial Inertial Instability STEPHEN D. GRIFFITHS Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom (Manuscript received 0 February 00, in final form 3 September 00) ABSTRACT The zonal flows of the earth s equatorial stratosphere and mesosphere are prone to inertial instability when the horizontal shear 0 at the equator. However, it is not clear why the vertical wavelength of the observed structures is much greater than that predicted by the simple linear theory, based on a vertical scale selection by molecular diffusion. Here, a nonlinear mechanism is described that can lead to upscaling from structures with short vertical wavelengths to long vertical wavelengths. The mechanism is dependent upon a secondary Kelvin Helmholtz instability that develops as the inertial instability grows. Numerical simulations on an equatorial plane, employing a simple parameterization of the vertical transport of horizontal momentum due to the secondary instability, are used to assess the likely degree of upscaling in a zonally symmetric system. For a fluid with buoyancy frequency N, it is shown that upscaling toward the buoyancy cutoff wavenumber 4N/ can occur, even when diffusion is so weak that the wavelength of the most unstable linear mode is much smaller than the buoyancy cutoff wavelength. Thus, the simulated latitudinal and vertical structure of the instability in the nonlinear regime is significantly different from that predicted by the linear theory, and can explain the scales of the structures observed near the equatorial stratopause. 1. Introduction Inertial instability can arise in rotating fluid systems when the magnitude of the absolute angular momentum decreases away from the rotation axis (e.g., Charney 1973). Such a configuration occurs in the equatorial stratosphere and mesosphere near solstice, where there is a strong and persistent cross-equatorial shear, so that the zonal wind increases moving into the winter hemisphere. Thus, sufficiently close to the equator, in the winter hemisphere, the magnitude of the absolute angular momentum decreases away from the rotation axis, and the flow may be prone to inertial instability. There are well-documented cases of structures near the equatorial stratopause, with a vertical wavelength of about 10 km, bearing a strong resemblance to unstable modes of the linear stability theory (e.g., Hitchman et al. 1987; Hayashi et al. 1998; Smith and Riese 1999). There seem to be several large-amplitude events of inertial instability per year. However, our understanding of these events remains incomplete. Some insight may be gained from the linear instability theory for a vertically uniform flow with constant latitudinal shear on an equatorial plane, as developed by Dunkerton (1981, 198, 1983). The theory predicts that the instability will take the form of over- Corresponding author address: Stephen D. Griffiths, Dept. of Atmospheric Sciences, University of Washington, Box , Seattle, WA sdg@atmos.washington.edu turning cells in the meridional plane, and that in the absence of diffusion the most unstable mode will have a vanishingly small vertical scale. However, taking account of the small molecular diffusivity near the stratopause, and other parameter values, such as the strength of the shear and the stratification, the linear theory predicts a preferred vertical wavelength of about 1 km. Despite the limitations of the theory, and the limited vertical resolution of the satellite-based observations of the instability, there is an unexpectedly large difference between the predicted vertical wavelength of about 1 km and the observed vertical wavelength of about 10 km. One possible explanation is that throughout the region of instability there exists some kind of background eddy diffusion, much larger than the molecular diffusion, that suppresses the smaller-scale structures. This scenario has been discussed by Hitchman et al. (1987, section 6b), on the basis of mixing induced by breaking gravity waves in the lower mesosphere, although the required values for the eddy diffusivity, of the order of 50 m s 1, are often regarded as being unrealistically large. Another possibility is that the zonal asymmetry of the basic state somehow leads to a larger vertical scale, perhaps as described by Clark and Haynes (1996), who studied the linear absolute instability of a zonally asymmetric flow. Here, we will study a distinct nonlinear process that could lead to vertical upscaling of the required degree. We consider inertial instability in the strongly unstable regime, in which diffusion is relatively weak, as is likely 003 American Meteorological Society

2 978 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 to be the case in the equatorial stratosphere and mesosphere near solstice. In this regime, the vertical shear layers produced by the evolving inertial instability are strong enough for a secondary Kelvin Helmholtz (KH) instability to occur. Since the high wavenumber, most inertially unstable modes induce the strongest vertical shears, it is possible that they will become inhibited by secondary Kelvin Helmholtz instabilities, while the low wavenumber, less inertially unstable modes will continue to grow. The low wavenumber modes could come to dominate the evolution of the instability, while the high wavenumber modes are continually suppressed. In effect, there would be a nonlinear vertical scale selection induced by the secondary instabilities, quite independent of the weak vertical diffusion. Indeed, even in the complete absence of vertical diffusion, this mechanism would lead to a finite vertical scale selection. For simplicity, we will study this process for a basic flow that is vertically uniform and with constant latitudinal shear, and we will take the Prandtl number to be unity. Furthermore, since the most unstable linear mode is zonally symmetric in this strongly unstable regime, we will restrict attention to the zonally symmetric instability throughout, hoping that this will adequately illustrate the upscaling process. We also neglect radiative damping since, as shown by Clark and Haynes (1994), the inclusion of Newtonian cooling does not lead to a finite linear vertical scale selection. The approach is as follows. In section, the formulation and theoretical background are outlined. The linear stability theory is briefly reviewed, and extended to derive a scaling for the most unstable vertical wavenumber selected by vertical diffusion. In section 3, we motivate and describe the nonlinear vertical scale selection mechanism more precisely. A numerical model is introduced to study the mechanism, including a parameterization of the vertical transport of horizontal momentum due to the secondary Kelvin Helmholtz instabilities. In section 4, we present results from this numerical model, enabling the viability of the nonlinear vertical scale selection mechanism to be assessed. In section 5, we discuss the nature of the associated dynamics and inertial equilibration. We conclude in section 6, by discussing how this study may improve our understanding of inertial instability in the equatorial stratosphere and mesosphere.. Formulation and background a. The governing equations We study the inertial instability of a stratified fluid on an unbounded equatorial plane. We use a set of Cartesian coordinates, with the x axis pointing eastward, the y axis pointing northward, and the z axis as a local vertical coordinate. We suppose that the flow is zonally symmetric (i.e., /x 0), hydrostatic, and that the diffusivities of heat and momentum take the same value. Furthermore, since the inertial instability is expected to occur in thin vertical layers, we use the Boussinesq approximation, and approximate the diffusion operator as /z. The complete set of equations is u u f t z u 1 p u fu t y z 00 p g z u t z w 0, (1) y z where u,, and w are the fluid flow speeds in the x, y, and z directions, respectively; is the density; 00 is a constant reference value for the density; p is the pressure; g is the acceleration due to gravity; and f y. We consider a basic flow u y, w 0, 00 0 (z), where N 1 g 00 0 /z is a positive constant. We take 0 for definiteness, so that the unstable region with an adverse gradient of angular momentum is 0 y /. We study the stability of this flow to zonally symmetric disturbances u,,, where is a streamfunction for the flow in the meridional plane; that is, /z, w /y. b. The linear stability problem The linear stability problem was originally studied by Dunkerton (1981). We will briefly review his results, and extend them to examine linear vertical scale selection by diffusion. We ignore the terms quadratic in disturbance amplitude in (1), and look for latitudinally bounded disturbances of the form Re{(y)e imzst }. Thus, Re(s) is the growth rate of the disturbances, and m is the vertical wavenumber, which we take to be positive without loss of generality. The linearized disturbance equation is readily seen to be of the Hermite type, and the requirement of latitudinal boundedness leads to solutions only when N (s m ) (n 1), () 4 m for any integer n 0. We immediately note that any unstable mode must satisfy 4N m ; (3) that is, there is a buoyancy cutoff wavenumber inde-

3 1APRIL 003 GRIFFITHS 979 pendent of diffusion. The cells cannot be too deep because of the work required to overturn the stratification surfaces. For the most unstable mode, corresponding to n 0 in (), 1/(Y/L) (imzst) Re{Ae e }, (4) 1/(Y/L) (imzst) Re{AYe e }, (5) for some complex number A, where Y y, (6a) 1/ N L. (6b) m Here, Y is a latitudinal coordinate shifted to the center of the unstable region, and L is the equatorial Rossby length. There is an overturning motion in vertically stacked cells of height /m, and of latitudinal scale L. In the regions of rising and sinking air, there are density perturbations, which are largest at Y L, namely, at y / L. Using (3) and (6b), we see that all growing modes have L /, so the maximum density perturbations both lie within the initially unstable region. The density perturbations give rise to the temperature perturbations observed in events of inertial instability in the equatorial atmosphere. c. Linear vertical scale selection Returning to (), we see that if 0, then the most unstable mode, with growth rate /, is found as m. In a real fluid there will be a range of processes that prevent structures forming on such vanishingly small scales. At the very least, molecular diffusion will become important, and hence it is essential to consider the case 0. Then viewed as a function of m, the growth rate has a single maximum s s *,atmm * say. Our next aim is to find expressions for s * and m *. We choose to work with nondimensional variables ŝ (/)s, and mˆ ( /N)m, where 1/3 N (7) 5 is a useful nondimensional parameter. With these choices, the eigenvalue relation () becomes 4 (ŝ mˆ ) 1 (n 1), mˆ for any integer n 0. To find the most unstable mode we set n 0, differentiate with respect to mˆ, and set ŝ/mˆ 0. After a little manipulation we find that the most unstable wavenumber mˆ * and the maximum growth rate ŝ * satisfy 6 5 mˆ * 4mˆ * 1, ŝ * mˆ * (mˆ * 5). (8a,b) Equation (8a) cannot be solved exactly for mˆ *, but, it is possible to make some useful progress. We are looking for solutions of (8a) for real positive mˆ *, and it is clear, graphically, that there is only one such solution, as expected. Further, we note from (8b) that there is instability ŝ * 0 mˆ * 5. Using (8a) once again, it is clear, graphically, that mˆ * Thus, there is instability if and only if 5/6 c (9) Dunkerton (1981) considered the marginal stability problem, and gave this result in a slightly different form. Thus, since is small in the case of instability, we may usefully write mˆ * as a power series in. Working from (8a), the first few terms are 10 mˆ * 1, (10) 3 9 which is accurate to within 6% for 0 c (Griffiths 000, p. 185). However, it is possible to say more. As can easily be verified from (8a), (i) mˆ * 1at0, (ii) mˆ */ 0 for 0 c, and (iii) mˆ * 5 1/ at c, so that mˆ * remains O(1) over the whole range of instability 0 c. Thus the most unstable (dimensional) wavenumber m * satisfies 1 N N m*, (11) the approximation becoming exact as 0. We can see the competition between stratification and the effect encouraging smaller vertical scales, and diffusion and cross-equatorial shear encouraging larger vertical scales. Note that if is increased by a factor of 8, then m * approximately halves. The most unstable wavenumber is relatively insensitive to the diffusivity. Thus we have described, via (11), the linear vertical scale selection by diffusion, for zonally symmetric modes. There also exists a body of theory for the zonally asymmetric modes of inertial instability on a uniform shear flow (Boyd and Christidis 198; Dunkerton 1983, 1993; Clark and Haynes 1996), revealing that the fastest growing mode need not be zonally symmetric. However, denoting the zonal wavenumber by k, the analysis of Clark and Haynes (1996) reveals that, for given m, the zonally symmetric mode is fastest growing as k 0 when m 4N / c, where c Numerical results, such as those shown in Fig. 1 of Dunkerton (1993), suggest that for given m, the zonally symmetric mode remains fastest growing for all k when m 4N c /. Thus, if m * 4N c /, we expect the fastest growing mode to be zonally symmetric. Using the approximation (10) for m *, this occurs when 0.1. This serves as a guide to when the fastest growing mode will be zonally symmetric, since there could be a faster growing zonally asymmetric mode with m 4N /. c 1/3

4 980 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 d. The stability parameter It is possible to nondimensionalize the nonlinear set of Eqs. (1) in such a way that the only parameter left in the system is just. Hence, in this zonally symmetric system with Prandtl number unity, the inertial instability is completely characterized by the parameter. One can see, from (7), that is a particular ratio of the stabilizing influences in the system diffusion, stratification, and the effect to the destabilizing influence of shear. Thus, is a complete measure of the inertial instability of the flow. When is not much less than the critical value for stability c 0.6, the flow is only weakly nonlinear, a regime that has been investigated analytically and numerically by Griffiths (003). In this regime the instability retains a regular form, and is easily understood. It seems unlikely that any vertical upscaling could occur here. The numerical simulations of Hua et al. (1997), for a nonhydrostatic system with Rayleigh friction, Newtonian cooling, and enhanced horizontal diffusion, indicated that some upscaling could occur in a moderately unstable regime, but the mechanism for their results is not clear. As decreases further, the inertial instability becomes more vigorous and the maximum linear growth rate s * increases towards the inviscid maximum /. By 0.0 (s * 0.9 /), the nonlinear evolution of the most unstable linear mode, in isolation, leads to regions where the Richardson number (Ri) falls to less than 0.5, near to maximum disturbance amplitude (Griffiths 003). For smaller still, the density contours overturn during the instability, leading to a secondary convective (i.e., gravitational) instability. In a numerical model unable to resolve these secondary instabilities, this quickly leads to features on the smallest scales, and unless these features are artificially suppressed, perhaps by hyperdiffusion, the numerical simulations will fail. The situation is increasingly complex and numerically demanding. The strongly unstable regime with K c has not been satisfactorily investigated. It is possible to estimate a typical value for in the equatorial stratosphere and mesosphere near solstice. Near the stratopause, N 0.0 s 1, and the molecular diffusivity m s 1. The study of Hitchman and Leovy (1986) indicates that during December and January, the cross-equatorial shear is 3 5 day 1 at 50 km, increasing to 4 7 day 1 at 70 km. In their observational study of inertial instability, Hayashi et al. (1998) estimate as s 1 (that is approximately 4 day 1 ) when the instability occurs. Thus, taking m 1 s 1, and in the range 3 6 day 1, (7) indicates that lies in the range , that is, near solstice, the upper stratosphere and lower mesosphere appear to be in the strongly unstable regime K c. From (11), the most unstable wavelength should be about 1 km. 3. Nonlinear vertical scale selection We now turn to consider in detail how inertial instability might develop in the strongly unstable regime. a. A secondary Kelvin Helmholtz instability According to the classical theory (Miles 1961; Howard 1961), a necessary condition for the linear instability of an inviscid, stratified, shear flow u u(z) is that the Richardson number N (z) 1 Ri (u/z) 4 somewhere in the fluid. The growth rate and preferred horizontal wavenumber of this Kelvin Helmholtz instability are expected to be of the order of the buoyancy frequency N and the inverse of the shear layer depth, respectively (e.g., Drazin and Reid 1981). The instability leads to an intense rolling up of the density contours, a process that is subject to a secondary convective instability, and various secondary three-dimensional instabilities (see Caulfield and Peltier 000, and references therein). Numerical simulations show that during KH instability, the Richardson number typically increases, sometimes to well above the critical value of 1/4 (e.g., Scinocca 1995; Fritts et al. 1996). For inertial instability dominated by a single vertical mode, both u and have sinusoidal dependence on z. As the amplitude of the instability increases, the magnitudes of u/z and /z increase, and, for 0.0, eventually the Richardson number locally falls to below 1/4 at some point during the evolution. Thus, a secondary KH instability is likely to occur, before density contour overturning. Since the KH instability occurs on a much faster timescale (of the order of N 1 ) than the inertial instability (of the order of 1 ), a KH instability would locally destroy the structure of the evolving inertial instability. The delicate balances required to create a coherent growing mode would be disrupted, and one would expect the growth rate of the inertial instability to decrease. b. The nonlinear vertical scale selection hypothesis However, so long as the inertial instability has not become sufficiently nonlinear to neutralize the mean flow (i.e., eliminate the cross-equatorial shear), the flow will remain unstable, and inertial instability will be driven. The situation becomes of special interest in the strongly unstable regime since, even if the most unstable mode is suppressed, there remains a wide range of unstable vertical wavenumbers. The unstable modes with wavenumbers greater than m * the most unstable wavenumber are unlikely to be of any interest, since their smaller vertical scales make them more susceptible to secondary KH instabilities. Thus, we are left to consider the unstable modes with wavenumbers less than m *.We

5 1APRIL 003 GRIFFITHS 981 FIG. 1. The nonlinear vertical scale selection hypothesis. The solid line is the linear growth rate Re(s) as a function of the vertical wavenumber m. anticipate that the low wavenumber modes close to the buoyancy cutoff would be able to evolve freely with no secondary instability. However, these modes are only weakly inertially unstable and, in isolation, are only able to grow to a small amplitude, as shown by the weakly nonlinear analysis of Griffiths (000, chapter 5). As the wavenumber increases toward m *, the modes become more inertially unstable and will have the potential to grow to larger amplitudes. Provided the increasing amplitudes and decreasing vertical scales do not lead to a secondary KH instability, it is possible these modes will become more prominent than the most unstable linear mode. The scenario is illustrated in Fig. 1. An analogous nonlinear selection mechanism for baroclinic instability in the atmosphere has been described by Welch and Tung (1998). In reality, the evolving inertial instability will comprise a whole range of vertical wavenumbers, and the situation will be considerably more complicated than the simple mode-by-mode interpretation just given. In particular, the KH instability occurs locally in physical space, so that modes of all vertical wavenumbers are affected by any single KH event. And, when the inertial instability becomes large enough, there will be considerable nonlinear interactions between the modes. However, it seems reasonable that some aspects of the behavior described above will feature in the inertial instability of strongly unstable flows. Then we might ask, which modes are sufficiently unstable, yet sufficiently unaffected by KH instabilities, to be able to grow to a large amplitude? In effect, which vertical scale does the instability select? What is the time evolution of the instability in this regime? And, a question of particular interest with observations of equatorial inertial instability in mind is, what is the appearance of the corresponding density perturbations? Some observational evidence that this sort of regime might occur in the real atmosphere is provided by Fritts et al. (199), who made a short study of the equatorial mesosphere near solstice using ground-based radar measurements with a high vertical resolution of less than 1 km. They observed structures with a vertical scale of 6 10 km, which they attributed to inertial instability. However, they also noted the simultaneous existence of low Richardson numbers, suggesting that KH instabilities might be dynamically important for these large amplitude structures. To study this nonlinear vertical scale selection mechanism by analytical means seems to be out of the question, so one must resort to numerical simulations. We can capture the essential features of inertial instability within a zonally symmetric model, thus greatly reducing the numerical complexity. However, the fastest growing KH instabilities will have a zonal dependence, and have preferred horizontal scales much smaller than those of the inertial instability. Thus, in a numerical model optimized to resolve the inertial instability, it will be impossible to satisfactorily resolve the KH instabilities. However, since the KH instabilities occur on a timescale much shorter than that of the inertial instability, it may be reasonable to parameterize their effects, even within a zonally symmetric model. c. The simulations We solve the set of Eqs. (1) numerically for the basic flow u 0 y. We use a pseudospectral model with a Fourier basis for both the exponentially decaying latitudinal structure, and the periodic vertical structure, as described in Griffiths (003). There is no explicit horizontal diffusion, other than that associated with the shear adjustment scheme, to be introduced below. The simulations were performed in the latitudinal domain 3.5 / y 4.5 /, at a resolution of 56 56, with a time step t 0.0 /. To enable the study of the whole range of initially unstable wavenumbers, the height of the computational domain was chosen to be somewhat larger than the buoyancy cutoff wavelength. The linear modes for this system can be generated numerically, showing excellent agreement with the theoretical predictions. Unless otherwise stated, each of the simulations presented below was initialized with the most unstable linear mode. The amplitude was chosen so that initially nonlinear terms were negligible, and the Richardson numbers were greater than d. Parameterizing the Kelvin Helmholtz instability As already discussed, it is necessary to parameterize the effects of the KH instabilities. The choice of parameterization scheme is a compromise between simplicity and the likely physical reality. Numerical results of KH instabilities, by Scinocca (1995) for instance, show the action of KH instability to be an increase in the local Richardson number, principally brought about by a reduction of the vertical shear in and around the initially unstable layer. Density is often partially mixed across the same region. With this in mind, the KH instabilities are parameterized using a vertical shear adjustment scheme, that simply reduces the vertical shear of u and where necessary, and hence increases the

6 98 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 FIG.. The vertical shear adjustment scheme. (left) An unadjusted profile U(z) as a solid line, with the two limiting profiles U 1 (z) and U (z) marked by solid disks and open diamonds respectively. (right) The adjusted profile U(z) as a thick line, with the unadjusted profile as a thin line. local Richardson number. For simplicity, no change is made to the density field. Thus, the scheme can do nothing about convective overturning, although the parameterization acts in such a way that convective overturning is unlikely to occur. The basis for the shear adjustment is a scheme given by Dunkerton (000), and illustrated in Fig.. Given a horizontal velocity profile U(z), to be adjusted around z z c, and an adjustment parameter, satisfying 0 1, the scheme adjusts U(z) so that the vertical shear U(z) atz z c is reduced to U(z c ). From the first height z z 1 above z z c where the vertical shear falls to U(z c ), a tangent is drawn. The first height below z z c where this tangent cuts U(z) is labeled z z. Similarly, from the first height z z 3 below z z c where the vertical shear falls to U(z c ), a tangent is drawn. The first height above z z c where this tangent cuts U(z) is labeled z z 4. Two limiting velocity profiles are formed for U(z) as follows: U(z) z z 1 U 1(z) U(z 1) U(z c)(z z 1) z z z1 U(z) z z U(z) z z 4 U (z) U(z 3) U(z c)(z z 3) z3 z z4 U(z) z z 3. The adjusted velocity profile U(z) is the unique linear combination of U 1 (z) and U (z) that conserves momentum. The scheme cannot lead to an increase in the shear at any height. The numerical implementation of the adjustment scheme is as follows. Every time step, the Richardson number N (z) Ri (u/z) (/z) is calculated throughout the domain. If it falls below some specified value Ri crit, then a shear adjustment is performed on u and about the point where Ri is lowest. The same adjustment parameter is used for adjusting both u and, and it is chosen so that the minimum Richardson number at that latitude becomes Ri crit. The Richardson number is recalculated throughout the domain, and a shear adjustment is performed again if there is another point at which Ri Ri crit. The procedure is repeated, if necessary, although for computational practicality a maximum number of adjustments per time step is specified. Since each shear adjustment is performed at a single latitude, lateral discontinuities will form within u and. Hence, it will be necessary to smooth out these fields, which we choose to do by applying some horizontal hyperdiffusion. Even though the hyperdiffusion will be most effectual where u and have the smallest scales that is, around the adjustment latitudes it is somewhat unsatisfactory to be applying horizontal hyperdiffusion to the whole domain. Hence, this should be done as sensitively as possible. First, no hyperdiffusion need be applied to the vertical mean of the velocity fields, since they are left untouched by the shear adjustment scheme. Second, the hyperdiffusion should be chosen to have a negligible direct effect on the scales of interest. For instance, the linear growth rate and latitudinal structure of the modes between the buoyancy cutoff and the most unstable wavenumber should be largely unaffected. For this study, it was found that a hyperdiffusion of the form 4 4 /y 4, with m 4 s 1, was satisfactory in these regards. Further, if the model is run with such a horizontal hyperdiffusion, but without the vertical shear adjustment scheme, then there is little or no vertical upscaling. Hence, the complete parameterization of the KH instabilities consists of a series of vertical shear adjustments, and the application of horizontal hyperdiffusion. No explicit vertical diffusion is applied during the adjustment. While the shear adjustment scheme does conserve horizontal momentum, it does not necessarily globally conserve potential vorticity, for instance. However, in the simulations to be presented, the scheme was in almost continual use, and the domain-integrated potential vorticity was typically conserved to within 0.005%. 4. Numerical results We will present results at three different values of the parameter (N / 5 ) 1/3. The simulations were performed at 4 day 1, N 0.0 s 1, and m 1 s 1, so that, as we vary, we are in effect varying. However, each simulation holds for other values of, N,, and, provided they are chosen so that is unchanged. a We first study simulations at 0.08, corresponding to 0.6 m s 1 at 4 day 1. The maximum

7 1APRIL 003 GRIFFITHS 983 FIG. 3. The time evolution at 0.08, with Ri crit 0.5. (left) The (nondimensional) disturbance amplitude D(t). (right) The evolution of the maximum deviation for each vertical Fourier component of the density. The horizontal dashed line denotes the buoyancy cutoff wavenumber. The field plotted is / 00, and the values have been nondimensionalized by N /g. Thus, at 4 day 1, a value of 0.1 corresponds to a fractional density perturbation / Taking a background temperature T K, this corresponds to a temperature perturbation of 4.9 K. (bottom right) The minimum Richardson number in the computational domain, Ri min. The time is expressed in units of /, soif4 day 1, then 1 time unit 0.5 days. growth rate s * 0.75 /, and the most unstable wavenumber m * is approximately 3 times greater than the buoyancy cutoff wavenumber. The domain height was /m 0, where the minimum resolved wavenumber m 0 was chosen to be m * /6. In the forthcoming discussion, the mode with vertical wavenumber n m 0 will be referred to as mode n. Thus, of the resolved modes, mode 1 has the largest wavelength, and mode 6 is the most unstable linear mode. The linear growth rates of modes 1 to 6, nondimensionalized by /, are 10 3, 0.99, 0.608, 0.700, 0.736, and Mode 1 is weakly decaying, mode is moderately unstable, while modes 3 6 are strongly unstable. Above mode 6, the growth rate decreases. We will also use the following nondimensional measure of disturbance amplitude: [ ] g m0 (y, z, t) D(t) dy dz, (1) N 00 where the integral is taken over the computational domain. Then, in the linear approximation, for a mode of the form (5) with vertical wavenumber m, the maximum slope max of the density contours will be related to D by 1/4 3/4 D m max. /N N For the first simulation, initialized with the most unstable linear mode, the shear adjustment scheme was applied, when necessary, with a critical Richardson number Ri crit 0.5. The time evolution of the disturbance amplitude D is shown in Fig. 3. After the initial exponential growth, there is an unsteady growth to a maximum disturbance amplitude, followed by a slow, unsteady decay. Throughout the simulation, the structure of the density perturbation appears to be predominantly mode 6, that is, on the scale of the most unstable wavenumber. However, it is possible to form a more precise measure of which modes dominate the evolution of the instability. Every few time steps, a vertical Fourier decomposition of the density perturbation is performed, and the maximum latitudinal value of each wavenumber component is recorded. The results of this wavenumber decomposition are also shown in Fig. 3. For simplicity the field has been plotted as if it were continuous in wavenumber space. However, the numerical model only resolves the discrete wavenumbers indicated, and only the values corresponding to these wavenumbers have any physical significance. The geometry of the domain restricts the possible vertical scales of the instability to a small discrete set. One can see that mode 6, the most unstable linear mode, is dominant during the early stages of the instability (in accordance with the initial conditions). Even though mode 4 does grow to a significant amplitude, mode 6 gives the largest density perturbation of any mode throughout the whole evolution for these initial conditions. Presumably mode 6 can become quite nonlinear before being suppressed, so that the mean flow has been fairly well neutralized before the other modes have grown to a comparable magnitude. Thus, for these parameters, there is little sign of upscaling. Also shown in Fig. 3 is the time evolution of Ri min, the minimum Richardson number in the computational domain. One can see that the shear adjustment starts shortly before t 50 /. Subsequently, Ri min is held very close to 0.5, before the disturbance starts to decay and Ri min increases. To assess the importance of the initial conditions, a simulation was performed starting from a state with all

8 984 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 FIG. 4. Results from the simulation at 0.08, with mixed-mode initial conditions and Ri crit 0.5. (left) The time evolution of the maximum deviation for each vertical Fourier component of the density, and the minimum Richardson number. The horizontal dashed line denotes the buoyancy cutoff wavenumber. (right) The density perturbation at t 30.6 /. Only the center of the latitudinal domain is shown (although the complete vertical extent is shown). The vertical dashed lines denote the boundaries of the initially unstable region. the linear modes of interest present. In particular, modes 4, 5, and 6 were initially dominant and of comparable amplitude. The same shear adjustment parameters were used as for the first simulation. The subsequent evolution of the maximum deviation for each vertical Fourier component of the density is shown on the left-hand side of Fig. 4. Note that the mode component, perhaps generated initially by an interaction between modes 4 and 6, is able to grow to a large amplitude before the flow has been neutralized. The subsequent large amplitude evolution is dominated by several modes, in particular modes 4 and 6, and thus the apparent vertical wavenumber of the instability would depend upon the time of observation. For instance, shown on the righthand side of Fig. 4 is the density perturbation at t 30.6 /. One can see the wide fringes of a mode disturbance, as well as disturbances of a higher wavenumber near the centre of the domain. Thus, parameterizing the effects of KH instabilities, several vertical modes can be important in the evolution of inertial instability at It appears that the most unstable linear mode will not be completely suppressed by the KH instabilities, and that it can grow to a significant amplitude. However, the effect of the initial conditions is great, and depending on how they are chosen the least unstable of the resolved growing modes (mode in these simulations) can come to have the largest amplitude at a given time. Some nonlinear vertical scale selection does occur, although it appears to be a complex process with more than one wavenumber involved. b We extend our understanding by studying simulations performed at 0.05, corresponding to m s 1 at 4 day 1. The maximum growth rate s * 0.84 /, and the most unstable wavenumber m * is approximately 5 times greater than the buoyancy cutoff wavenumber. The minimum resolved wavenumber m 0 was chosen to be m * /10, so that mode 10 is the fasting growing linear mode. Mode 1 is weakly decaying, mode is moderately unstable with a growth rate nondimensionalized by / of 0.185, mode 3 has a nondimensional growth rate of 0.59, while modes 4 to 10 have nondimensional growth rates between and the maximum value of We set Ri crit 0.3, and once again take initial conditions with only the most unstable mode present. The resulting time evolution is shown on the left-hand side of Fig. 5. One can see that mode 10 is the largest mode initially, as expected, but that subsequently modes, 3, and 4 become dominant. One can also see that Ri min is held around 0.5, with occasional brief excursions to lower values. Note that the data has been smoothed so that higher frequency variations of Ri min are masked. For these parameters the adjustment process is not perfect recall that Ri crit 0.3 but by choosing Ri crit just a little larger than 0.5, we obtain the desired behavior. Shown in Fig. 6 is the density perturbation at two different times one early in the evolution, and one near to maximum disturbance amplitude. The vertical upscaling is evident. Also note that at maximum disturbance amplitude the sets of density perturbation extrema are at y 0 and y 1.4 /. Thus, the density extrema have drifted away from their starting positions near the center of the initially unstable region. This behavior is reminiscent of the diffusive weakly nonlinear regime (see the discussion in Griffiths 003). Furthermore, as pointed out in section b, the linear modes for a uniform shear flow necessarily have both density extrema inside the initially unstable region, that is, between the dashed lines in Fig. 6. Thus, since nearto-maximum disturbance amplitude the density extrema

9 1APRIL 003 GRIFFITHS 985 FIG. 5. The time evolution at Shown are the maximum deviation for each Fourier component of the density, and the (smoothed) minimum Richardson number. (left) The simulation with Ri crit 0.3; (right) the simulation with Ri crit 0.4. The horizontal dashed line denotes the buoyancy cutoff wavenumber. lie outside the initially unstable region, their structure is a result of nonlinear behavior. To assess the sensitivity of the results to the parameters of the shear adjustment scheme, a simulation was performed with the same initial conditions but taking Ri crit 0.4. The time evolution is shown on the righthand side of Fig. 5. The shear adjustment scheme succeeds in keeping Ri min between 0.5 and 0.4, with occasional excursions to lower values. The evolution of the maximum deviation for each vertical Fourier component of the density is to be compared with the plot to its left, from the simulation with Ri crit 0.3. Although, quantitatively, there are differences between the two simulations, qualitatively there are great similarities. In particular, the upscaling to a large amplitude phase dominated by modes 3 and 4 appears to be robust. c Further simulations were performed at 0.04, corresponding to 0.03 m s 1 at 4 day 1. The maximum growth rate s * 0.88 /, and the most unstable wavenumber m * is approximately 6 times greater than the buoyancy cutoff wavenumber. The minimum resolved wavenumber m 0 was chosen to be m * /1. Mode 1 is weakly decaying, mode is moderately unstable with a growth rate nondimensionalized by / of 0.57, mode 3 has a nondimensional growth rate of 0.61, while modes 4 to 1 have nondimensional growth rates between 0.76 and the maximum value of The time evolution, with Ri crit 0.3, is shown on the left-hand side of Fig. 7. One can see that mode 1 is the largest mode initially, as expected, but that, subsequently, mode 3 becomes dominant. Once again, even though the shear adjustment scheme is struggling to consistently maintain Ri min 0.3, the Richardson number is adequately limited throughout the simulation. Shown in Fig. 8 is the density perturbation at two different times during the evolution, providing another clear example of upscaling. Once again, at maximum disturbance amplitude, the density perturbation extrema FIG. 6. The density perturbation at two different times during the simulation at 0.05, with Ri crit 0.3: (left) at t 6 /, shortly after nonlinear effects become important; (right) at t 59. /, near to maximum disturbance amplitude. The vertical dashed lines denote the boundaries of the initially unstable region.

10 986 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 FIG. 7. Results from the simulation at 0.04, with Ri crit 0.3. (left) The time evolution of the maximum deviation for each vertical Fourier component of the density, and the (smoothed) minimum Richardson number. The horizontal dashed line denotes the buoyancy cutoff wavenumber. Right is the density perturbation at t 40 /. The vertical dashed lines denote the boundaries of the initially unstable region. have drifted away from their starting positions within the initially unstable region. Also evident from Fig. 7 is the generation of a persistent mode 1 disturbance after the large amplitude phase of the instability. Shown on the right-hand side of Fig. 7 is the density perturbation at t 40 /, clearly revealing this structure. The values of / 00 associated with it, approximately 0.04 N /g, correspond to a temperature perturbation of approximately K, taking 4 day 1 and a background reference temperature T K. The generation of such temperature perturbations, on a vertical scale larger than the buoyancy cutoff scale, is especially interesting from an observational perspective. However, even though the vertical upscaling near maximum disturbance amplitude is not sensitive to changes in the parameters of the shear adjustment scheme, the magnitude of this persistent mode 1 disturbance is sensitive to such changes. For instance, as one can see in Fig. 5, for the simulations at 0.05 with Ri crit 0.3 and Ri crit 0.4, the magnitude of the persistent mode 1 disturbance differs by a factor of. Thus, we shall merely note the generation of this mode, and comment no further upon its magnitude or structure. 5. The nature of the strongly unstable regime To interpret the dynamics of this strongly unstable regime, we will study the evolution of the vertical mean of the potential vorticity Q(y, t). The potential vorticity (PV) has been defined by ( f ez u) Q, (13) 0/z where the denominator 0 /z is a constant included simply for convenience. In particular, we will be interested in the quantity f Q, which determines the strength of inertial instability, when fq 0. Thus, to form an FIG. 8. The density perturbation at two different times during the simulation at 0.04, with Ri crit 0.3: (left) at t 4.8 /, shortly after nonlinear effects become important; (right) at t 30 /, near to maximum disturbance amplitude. The vertical dashed lines denote the boundaries of the initially unstable region.

11 1APRIL 003 GRIFFITHS 987 FIG. 9. Mean flow profiles at 0.05, with Ri crit 0.3. (top) The mean PV and the quantity fq at t 60 /, at maximum disturbance amplitude. (middle) The corresponding profiles at t 80.3 /, when (bttom) The corresponding profiles at t 00 /, near the end of the evolution. In each plot the dashed line is the initial profile, and the vertical dashed lines denote the boundaries of the initially unstable region. Only the center of the computational domain is shown. idea of the extent to which a mean flow has been neutralized, we introduce a neutralization parameter 0 (t) 1 fq dy fq dy, (14) U U 0 where Q 0 (y) is the vertical mean of the potential vorticity at t 0, U is the range of y for which fq 0 at any given time, and U 0 is the range of y for which fq 0 0, that is, 0 y /. Initially 0, and for a perfectly neutralized flow with fq 0 everywhere, 1. a. The mean flow change Shown in Fig. 9 are Q and f Q, at three different times from the simulation at 0.05 with Ri c 0.3. The top panel is taken at t 60 /, near maximum disturbance amplitude, when 0.49, while the middle panel is taken at t 80.3 /, when As discussed by Griffiths (003), in the weakly nonlinear regime the action of inertial instability is to approximately homogenize fq to a small negative value, in and around the initially unstable region. In this strongly unstable regime we see that the mean PV is changing to effect a similar homogenization of f Q. Shown in the bottom panel of the same figure are the corresponding profiles from well into the decay phase of the instability. Now fq has been homogenized to a very small negative value around the initially unstable region, presumably corresponding to a completely, or almost completely, neutralized flow, with Just as in the weakly nonlinear regime, the line of zero mean PV moves poleward during the nonlinear evolution of the instability, not equatorward. Note that, even though some horizontal hyperdiffusion has been applied to u and via the shear adjustment scheme, none has been applied to the vertical mean of these fields. Thus, the final mean PV profiles are not directly affected by horizontal hyperdiffusion. Also note that in three-dimensional reality, the reversed PV gradient near y 0.4 / would most likely be eliminated by barotropic instability. Since the nonlinear evolution is dominated by modes with a comparatively large vertical wavelength, one might be tempted to think that the dynamics are analogous to the diffusive weakly nonlinear regime, in which vertical diffusion suppresses the short wave length modes. In such a regime, diffusion can stabilize regions in which fq 0 to a certain degree, and the dynamics are driven only by the need to partially reduce the maximum value of ( f Q) (Griffiths 003). How-

12 988 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME TABLE 1. Neutralization times t neut. Run Ri crit 0.5 Ri crit 0.5 (mixed) Ri crit 0.3 Ri crit 0.4 Ri crit 0.3 t(ri min 0) (/) t( 0.75) (/) t neut (/) ever, as clearly illustrated in Fig. 9, in this strongly nonlinear regime the dynamics are indeed driven by the need to reduce ( f Q) to a very small value, throughout the initially unstable region. Thus, the dynamics, and the associated mean flow change induced by the instability, are quite different from those of the diffusive weakly nonlinear regime. b. The neutralization time How long does the inertial instability take to neutralize the mean flow? We estimate this as the elapsed time between the inertial instability becoming measurable, which we take to be when Ri min 0 (typically corresponding to a nondimensional disturbance amplitude D 0.001), and the time when the flow becomes almost neutralized, which we take to be when 0.75 (typically corresponding to a mean flow like that shown in the middle panel of Fig. 9). The corresponding intervals and neutralization times for each of the simulations are given in Table 1. One can see that neutralization times are approximately /. Thus, since the most unstable mode has an e-folding timescale of the order of /, the neutralization process takes rather a long time the secondary KH instabilities render the inertial instability rather slow at neutralizing the mean flow. This is presumably partly due to the fact that the low wavenumber modes have growth rates considerably smaller than /. As can be seen from Figs. 4, 5, and 7, within this large amplitude phase of the instability, the large amplitude disturbances at a given wavenumber tend to occur as a series of pulses, each pulse typically lasting about 10 /. Taking 4 day 1, each pulse would last about 5 days, and the large amplitude behavior would last about 0 30 days. c. The strongly unstable regime An interpretation of the strongly unstable regime, say for 0.05, is as follows. The most unstable linear mode is unable to grow to a large amplitude before becoming suppressed by secondary KH instabilities, and hence does not neutralize the mean flow completely with respect to disturbances of all wavenumbers. Thus, other modes, with lower wavenumbers, become important. Exactly which wavenumbers become important will depend upon the degree to which the mean flow has already been changed. The lower-wavenumber modes may themselves make a permanent mean flow change, but as suggested by the weakly nonlinear analysis of Griffiths (000, chapter 5) for modes close to the buoyancy cutoff, such modes will not necessarily neutralize the mean flow with respect to disturbances of a higher wavenumber. However, the simulations indicate that the combined action of the lower-wavenumber modes and the suppressed higher-wavenumber modes somehow neutralizes the mean flow completely. On the basis of such a complete neutralization of the mean flow, one can argue for a nonlinear vertical scale selection on the order of the buoyancy cutoff. One starts by asserting that the mean flow can only be neutralized when fluid parcels have moved across the entire unstable region. Since angular momentum is conserved, velocity amplitudes must become / (the width / of the unstable region, multiplied by the angular momentum gradient ). If the vertical wavenumber of the instability is m, this implies vertical shears m /, and hence Richardson numbers N /m 4. For the Richardson number to remain larger than 1/4 during the instability, we need m N/. Therefore, disturbances with a vertical scale substantially less than the buoyancy cutoff scale will inevitably cause the Richardson number to drop below 1/4 well before the instability can saturate, and only disturbances with a vertical scale on the same order as the buoyancy cutoff scale can allow the instability to saturate while remaining KH stable. 6. Conclusions We have studied zonally symmetric equatorial inertial instability in the strongly unstable regime, paying particular attention to a nonlinear vertical scale selection mechanism induced by secondary Kelvin Helmholtz (KH) instabilities. Although the investigation has been focused on the instability of a flow with uniform shear, the results are expected to be applicable to the inertial instability of a whole range of simple shear flows, due to the essentially local nature of inertial instability. Further, we would also expect the nonlinear vertical scale selection mechanism to be important in zonally asymmetric inertial instability, such as that generated by propagating Rossby waves. While not wishing to discount other possible dynamical effects, the simulations presented here strongly suggest that secondary KH instabilities will play a dominant role in the nonlinear evolution of inertial instability when the instability parameter (N / 5 ) 1/ The KH instability produces an upscaling in physical space, so that the inertial instability is not necessarily dominated by structures on the vertical scale of the most unstable linear mode. For 0.08, it appears that the evolution will be sensitive to the initial conditions, and that modes on the scale of the most unstable linear mode can be dominant. For 0.05, there are clear signs that the dominant mode will be well sepa-