The 2-D version of the non-hydrostatic fully compressible model MOLOCH developed at ISAC-CNR was used in idealized set-up to study the start-up and finite amplitude evolution of symmetric instability. The unstable basic state was designed by numerical integration of the equation which defines saturated equivalent potential vorticity q&lt;sub&gt;e&lt;/sub&gt;&lt;sup&gt;*&lt;/sup&gt;. We present the structure and growth rates of the linear modes both for a supersaturated initial state (&quot;super&quot;-linear mode) and for a saturated one (&quot;pseudo&quot;-linear mode) and the modifications induced on the base state by their finite amplitude evolution

M. Fantini

P. Malguzzi

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Adv. Geosci., 17, 1–4, 2008www.adv-geosci.net/17/1/2008/© Author(s) 2008. This work is distributed underthe Creative Commons Attribution 3.0 License.Advances inGeosciencesNumerical study of two-dimensional moist symmetric instabilityM. Fantini and P. MalguzziISAC-CNR, via Gobetti 101, 40129 Bologna, ItalyReceived: 15 October 2007 – Revised: 7 March 2008 – Accepted: 15 May 2008 – Published: 20 June 2008Abstract. The 2-D version of the non-hydrostatic fully com-pressible model MOLOCH developed at ISAC-CNR wasused in idealized set-up to study the start-up and ﬁnite am-plitudeevolution ofsymmetricinstability. The unstablebasicstate was designed by numerical integration of the equationwhich deﬁnes saturated equivalent potential vorticity q∗e. Wepresent the structure and growth rates of the linear modesboth for a supersaturated initial state (“super”-linear mode)and for a saturated one (“pseudo”-linear mode) and the mod-iﬁcations induced on the base state by their ﬁnite amplitudeevolution.1 IntroductionIn ordinary, stably stratiﬁed, atmospheric conditions the ﬁeldof potential temperature θ increases with height. If for anyreason (e.g. a warm bubble) the vertical gradient changessign, convective instability arises. The same principle can beformally stated for inertial instability in the horizontal direc-tion by deﬁning an angular momentum function M=v+f0x,where v is the geostrophic wind, supposed locally parallel,and x an horizontal coordinate across the wind, with arbi-trary origin.If we consider the two ﬁelds together, as we are boundto do, as they are in thermal wind balance, instability cantake place even when the two ﬁelds, θ and M, are separatelystable. This happens in the presence of a horizontal thermalgradient, and its associated vertical shear of the wind. If theslope of isentropic surfaces becomes steeper than the slopeof angular momentum surfaces, i.e. ifdzdx  θ=const≡ −∂θ/∂x∂θ/∂z>dzdx  M=const≡ −∂M/∂x∂M/∂z(1)then unstable motion can occur.It is easily seen that the above condition can be writtenJ (M,θ) < 0 (2)Correspondence to: M. Fantini(m.fantini@isac.cnr.it)for instability. In two dimensions this can also be expressedas the requirement that potential vorticity is negative: q<0.The above considerations hold for dry air. If moisture isto be considered, one can simply replace potential tempera-ture θ with the saturated equivalent potential temperature θ∗e .Then, if the air is saturated, an atmosphere withq∗e = J M,θ∗e< 0 (3)is unstable to inﬁnitesimal perturbations (linear instability).If the air is sub-saturated, one will need a ﬁnite amplitudeperturbation to lift air to its saturation level (conditional in-stability). Symmetric instability is often referred to as slant-wise convection, although studies of the energy conversionmechanism usually conclude that the release of gravitationalpotentialenergydoesnotplayamajorroleinitsdevelopment(e.g. Thorpe and Rotunno, 1989).A complete list of references on the instability, either dryor moist, inﬁnitesimal or ﬁnite amplitude, can be found in thereview by Schultz and Schumacher (1999). As far as mete-orological applications are concerned, symmetric instabilityis often invoked as a mechanism for the formation of frontalrainbands (e.g., Bennetts and Hoskins, 1979). An unrelatedphenomenon for which symmetric instability should be re-sponsible is the appearance of areas of zero equivalent poten-tial vorticity (i.e. neutral to symmetric instability) in the coreof severe midlatitude storms – often referred to as slantwiseconvective adjustment (see Emanuel, 1988; Kuo and Reed,1988). Finally the matter of parameterization of slantwiseconvection in numerical models was examined by Nordeng(1987) and Lindstrom and Nordeng (1992) with seeminglygood effects on the overall simulation of convection and pre-cipitation.2 ModelThe model we use for this study is a 2-D version of the non-hydrostatic fully compressible model MOLOCH, designed atISAC-CNR in recent years. The main characteristics of thissimpliﬁed version are that it integrates the fully compress-ible set of equations, using as prognostic variables pressure,Published by Copernicus Publications on behalf of the European Geosciences Union.2 M. Fantini and P. Malguzzi: Two-dimensional non-hydrostatic symmetric instabilityFig. 1. Potential temperature θ (color and thin contours every 10 K)and meridional wind v (thick contours every 10ms−1) for the basicstate designed for this study (see Sect. 3). The horizontal domainis covered by 1200 grid points, for a resolution of approximately2.2km.Fig. 2. Saturated equivalent potential temperature θ∗e (color andthin contours every 10K) and angular momentum M (thick con-tours every 20ms−1). The origin is arbitrary and here chosen forsymmetry. At the bottom is the horizontal proﬁle of saturated equiv-alent potential vorticity as from Eq. (7), which is used to design thebase state up to the tropopause (see text) .temperature, speciﬁc humidity, horizontal and vertical veloc-ity components, and ﬁve water species. Model dynamicsare integrated in time with an implicit scheme for the ver-tical propagation of sound waves, while explicit, time-splitschemes are implemented for the remaining terms. Advec-tion is computed using the Eulerian WAF scheme (Billet andToro, 1997). The vertical grid is stretched exponentially withheight.Horizontal fourth order diffusion and divergence dampingare included to prevent energy accumulation on the shorterspace scales. The physical scheme of the simpliﬁed versionconsists of only microphysics, based on the parameterisationproposed by Drofa and Malguzzi (2004). The physical pro-cesses determining the time tendency of speciﬁc humidity,cloud water/ice and precipitating water/ice are divided into”fast” and ”slow” ones. Fast processes involve transforma-tions between speciﬁc humidity and cloud quantities, whileslow ones involve rain/snow/hail production and fall. Tem-perature is updated by imposing exact enthalpy conservationat constant pressure. Fall of precipitation is computed withthe stable and dispersive backward-upstream scheme, withfall velocities depending on concentration.A fuller description of MOLOCH can be found in Mal-guzzi et al. (2006).3 Basic stateThe deﬁnition of saturated equivalent potential vorticityQ =−θ∗exvz + (vx + f0)θ∗ezα (4)(notice that the non-hydrostatic contributions to relative vor-ticity wx and wy disappear in the two-dimensional version,while the compressibility effects remain in the speciﬁc vol-ume α) was used as an equation for the vertical derivative ofθ∗e .Starting from an imposed temperature proﬁle at the sur-face T=T(x) at z=0 and a uniform pressure p=p0 we deﬁnepressure at the next upward level by integration of the hydro-static relation. Then the wind at the upper level is known bythe geostrophic relation so that (4) can be used to ﬁnd θ∗e atthe next level upward. At this point the deﬁnitions of equiva-lent saturated potential temperature θ∗e and saturation mixingratio qsatVθ∗e = θ∗e θ (T,p),qsatV (T,p)(5)are inverted numerically, with a simple Newton iteration, toﬁnd temperature TThis procedure is repeated for a chosen negative Q, andwith an extremely high vertical resolution (of the order of afew meters), upward to a predeﬁned “tropopause” level, afterwhich a very stable stratiﬁcation is imposed.Figures 1 and 2 show the base state adopted for the ex-periment that will be discussed in the next section. It wasgenerated with the above procedure starting from a surfacetemperatureT(x) = T0 + (1T)(1 − cos(x))2(6)with T0=293.15K (this is the temperature at the westernboundary) and 1T=10K (this is the thermal gradient overhalf the domain – each half of the domain is the mirror im-age of the other one, so that we were able to use periodicboundary conditions in the numerical integration).Adv. Geosci., 17, 1–4, 2008 www.adv-geosci.net/17/1/2008/M. Fantini and P. Malguzzi: Two-dimensional non-hydrostatic symmetric instability 3Fig. 3. Left: Vertical velocity component w (color: gray w<0, redw>0; contours every 1.0cms−1) at 17h after initialization, on asubdomain of Fig. 1. Right: Same except at 26h.Fig. 4. Left: Vertical velocity component w (color: gray w<0, redw>0; contours every 2.0cms−1 at 33h after initialization. Right:Saturated equivalent potential temperature θ∗e (color and thin con-tours every 10K) and absolute momentum M=v+f0x (thick con-tours every 10ms−1) at 33h.A horizontal modulation of Q was also imposed, to con-ﬁne the instability and avoid the occurrence of convectiveinstability.Q(x) = QMIN(1 − cos(2x))2(7)with QMIN= − 0.25PV units (1PVU is equivalent to10−6 MKS units).4 Reference experimentWe present here one run of the model, which was started withrandom noise of very small amplitude, and the most unstableperturbation developing was followed up to ﬁnite amplitude,when it is able to modify the base state. In order to show inone run all the features we discovered in the evolution of thisform of instability, we included an artiﬁcial amount of liquidwaterintheinitialcondition(twopercentofsupersaturation).In other words the domain is ﬁlled by a dense cloud, in whichenvironment the symmetric instability develops.Figure 3 shows perturbation ﬁelds of u and w at 17 and26hafterinitialization. Onlyabouthalfofthedomain, ontheFig. 5. Same as Fig. 4 except at 40h. w contours every 10cms−1.Atthisstageitcanbeseenhowuprightconvectioncrossesverticallythe M contours instead of nearly following them as in the previousﬁgure.left of the center of Figs. 1 and 2, is shown. At 17h the mostunstable mode is seen to have a roughly sinusoidal structureroughlyalignedwiththeM surfaces. Thisisduetothenearlyperfect symmetry of upward and downward motions in this“cloudy” environment. While the updrafts are accelerated byrelease of latent heat the downdrafts are equally acceleratedby reevaporation of the liquid water present in the environ-ment. This is a linear stage of the perturbation (what we call“super”-linear), which is of small amplitude, sinusoidal, andgrows exponentially (also see Fig. 6 below).After a while (Fig. 3 right panel) the reevaporating down-drafts exhaust all the liquid water present in the initial cloud,and the perturbation is beginning to assume the aspect of the“pseudo”-linear mode, with updrafts becoming stronger thandowndrafts. At this stage the perturbation is still small am-plitude, and grows exponentially, but does not possess all theproperties that deﬁne a linear mode, in particular there is nosuperposition principle for these solutions.Later on (Fig. 4) the “pseudo”-linear mode has completelytaken over, and has grown to an amplitude at which it inducesa visible modiﬁcation on the base state. From the θ∗e , M plot(Fig. 4 right) it can be seen that the effect of the perturbationhas been to move warm air aloft, and that in some regionsin mid-troposphere the vertical gradient of θ∗e changes sign.This is a situation unstable to vertical convection, and indeed(Fig. 5) this develops, on a faster time scale than the symmet-ric one.Figure 6 displays the growth rate computed every hourduring the run, where the two distinct exponential phases(where the growth rate is approximately a constant) are ev-ident, followed by the faster growth of upright convection,and ﬁnal nonlinear decay of the perturbation.Finally, in Fig. 7, we show the growth rates of the pseudolinear mode, as determined by numerical experiments, vs thepotential vorticity QMIN used in Eq. (7) to design the basestate. The growth rate is computed asσ =11tlogwMAX(t + 1t)wMAX(t)(8)www.adv-geosci.net/17/1/2008/ Adv. Geosci., 17, 1–4, 20084 M. Fantini and P. Malguzzi: Two-dimensional non-hydrostatic symmetric instabilityFig. 6. Growth rate σ of the perturbation during the reference run,as deﬁned in Eq. (8). The dashed line marks the growth rate of the“pseudo”-linear mode (see text)Fig. 7. Growth rate of pseudo-linear mode vs QMIN for cosineproﬁle as determined from the numerical model.For a comparison with baroclinic instability consider that themaximum Eady growth rate at Q=0, which coincides withRi=1, is 0.31f0. However, a typical Richardson number forsynoptic scales is of order 100, so that a more reasonable es-timate for the growth rate of baroclinic waves in midlatitudesis an order of magnitude smaller. For the dry equivalent ofFig. 7, computed analytically, see Fig. 4 of Stone (1966).All linear growth rates in Figs. 6 and 7 were obtained innumerical experiments initialized with random noise, andwith the amplitude of the perturbation reduced every timeit reached a predetermined threshold, therefore always main-taining it in the linear regime.5 Concluding remarksWe have presented in this work a few preliminary results inregard to two dimensional non hydrostatic numerical simula-tions of symmetric instability in a saturated environment.We have designed a basic state with prescribed negativepotential vorticity, in such a way as to avoid upright convec-tive instability while allowing symmetric instability to takeplace.In this environment we have identiﬁed two types of linearinstability, characterized, the ﬁrst (“super”-linear) by sym-metric (up- and downdrafts) structure and higher growth rate,and the second (“pseudo”-linear) by stronger updrafts andweaker downdrafts. The “super”-linear mode was obtainedby providing the initial state with enough liquid water tomaintain saturation by re-evaporation in the downdraft. Inthe following non linear stage symmetric instability is ableto trigger mid-level convection by “slantwise” advection ofwarmer air.For more details on the linear stages of development seeFantini and Malguzzi (2008), where we identify and studymore accurately the two types of “linear” modes, in regardto growth rates and their dependence on environmentalparameters and microphysical processes.Edited by: A. MugnaiReviewed by: R. RotunnoReferencesBennetts, D. A. and Hoskins, B. J.: Conditional Symmetric Insta-bility – A Possible Explanation for Frontal Rainbands, Q. J. R.Meteor. Soc., 105, 945–962, 1979.Billet, S. and Toro, E. F.: On WAF-type schemes for multidimen-sional hyperbolic conservation laws, J. Comput. Phys., 130, 1–24, 1997.Drofa, O. V. and Malguzzi, P.: Parameterisation of microphysicalprocesses in a non-hydrostatic prediction model, in: Proceed-ings of 14th Intern. Conf. on Clouds and Precipitation (ICCP),Bologna, Italy, 19–23 July 2004, 1297–1300, 2004.Emanuel, K. A.: Observational Evidence of Slantwise ConvectiveAdjustment, Mon. Weather Rev., 116, 1805–1816, 1988Fantini, M. and Malguzzi, P.: The slope of moist symmetricinstability with water loading, J. Atmos. Sci., 68, in press,doi:10.1175/2008JAS2695, 2008.Lindstrom, S. S. and Nordeng, T. E.: Parameterized slantwise con-vection in a numerical model, Mon. Weather Rev., 120, 742–756,1992.Kuo, Y.-H. and Reed, R. J.: Numerical simulation of an explosivelydeepening cyclone in the Eastern Paciﬁc, Mon. Weather Rev.,116, 2081–2105, 1988Malguzzi, P., Grossi, G., Buzzi, A., Ranzi, R., and Buizza,R.: The 1966 “century” ﬂood in Italy: A meteorologicaland hydrological revisitation, J. Geophys. Res., 111, D24106,doi:10.1029/2006JD007111, 2006.Nordeng, T. E.: The effect of vertical and slantwise convection onthe simulation of polar lows, Tellus, 39A, 354–375, 1987.Schultz, D. M. and Schumacher, P. N.: Use and Misuse of Con-ditional Symmetric Instability, Mon. Weather Rev., 127, 2709–2732, 1999.Stone, P. H.: On non-geostrophic baroclinic stability, J. Atmos. Sci.,23, 390–400, 1966.Thorpe, A. J. and Rotunno, R.: Nonlinear aspects of symmetricinstability, J. Atmos. Sci., 46, 1285–1299, 1989.Adv. Geosci., 17, 1–4, 2008 www.adv-geosci.net/17/1/2008/

Numerical study of two-dimensional moist symmetric instability

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