Theoretical and Computational Fluid Dynamics

, Volume 20, Issue 5–6, pp 377–404

# The Structure of Precipitation Fronts for Finite Relaxation Time

Original Article

## Abstract

When convection is parameterized in an atmospheric circulation model, what types of waves are supported by the parameterization? Several studies have addressed this question by finding the linear waves of simplified tropical climate models with convective parameterizations. In this paper’s simplified tropical climate model, convection is parameterized by a nonlinear precipitation term, and the nonlinearity gives rise to precipitation front solutions. Precipitation fronts are solutions where the spatial domain is divided into two regions, and the precipitation (and other model variables) changes abruptly at the boundary of the two regions. In one region the water vapor is below saturation and there is no precipitation, and in the other region the water vapor is above saturation level and precipitation is nonzero. The boundary between the two regions is a free boundary that moves at a constant speed. It is shown that only certain front speeds are allowed. The three types of fronts that exist for this model are drying fronts, slow moistening fronts, and fast moistening fronts. Both types of moistening fronts violate Lax’s stability criterion, but they are robustly realizable in numerical experiments that use finite relaxation times. Remarkably, here it is shown that all three types of fronts are robustly realizable analytically for finite relaxation time. All three types of fronts may be physically unreasonable if the front spans an unrealistically large physical distance; this depends on various model parameters, which are investigated below. From the viewpoint of applied mathematics, these model equations exhibit novel phenomena as well as features in common with the established applied mathematical theories of relaxation limits for conservation laws and waves in reacting gas flows.

### Keywords

Tropical atmospheric dynamics Tropical convection Moisture Nonlinear relaxation equations Hyperbolic free boundary problems

### PACS

92.60.Ox 92.60.Jq

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### References

1. 1.
Arakawa A., Schubert W.H.(1974). Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci. 31, 674–701
2. 2.
Betts A.K., Miller M.J.(1986). A new convective adjustment scheme. Part II: Single column tests using GATE wave, BOMEX, and arctic airmass data sets. Q. J. R. Meteor. Soc. 112, 693–709
3. 3.
Bourlioux A., Majda A.J.(1995). Theoretical and numerical structure of unstable detonations. Philos. Trans. R. Soc. Lond. A 350, 29–68
4. 4.
Bretherton C.S., Peters M.E., Back L.E.(2004). Relationships between water vapor path and precipitation over the tropical oceans. J. Clim. 15, 2907–2920
5. 5.
Chen G.Q., Levermore C.D., Liu T.P.(1994). Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47(6), 787–830
6. 6.
Colella P., Majda A., Roytburd V.(1986). Theoretical and numerical structure for reacting shock waves. SIAM J. Sci. Stat. Comput. 7(4), 1059–1080
7. 7.
Emanuel K.A.(1987). An air–sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci. 44, 2324–2340
8. 8.
Emanuel K.A.(1994). Atmospheric convection. Oxford University Press, LondonGoogle Scholar
9. 9.
Emanuel K.A., Neelin J.D., Bretherton C.S.(1994). On large-scale circulations in convecting atmospheres. Q. J. R. Meteor. Soc. 120(519): 1111–1143
10. 10.
Folland G.B.(1976). Introduction to partial differential equations. Princeton University Press, Princeton
11. 11.
Frierson D.M.W., Majda A.J., Pauluis O.M.(2004). Large scale dynamics of precipitation fronts in the tropical atmosphere: a novel relaxation limit. Commun. Math. Sci. 2(4): 591–626
12. 12.
Fuchs E., Raymond D.J.(2002). Large-Scale Modes of a Nonrotating Atmosphere with water vapor and cloud-radiation feedbacks. J. Atmos. Sci. 59, 1669–1679
13. 13.
Gill A.E.(1986). Atmosphere–ocean dynamics, International Geophysics Series.vol. 30. Academic, New YorkGoogle Scholar
14. 14.
Harten A., Engquist B., Osher S., Chakravarthy S.R.(1987). Uniformly high-order accurate essentially nonoscillatory schemes. III. J. Comput. Phys. 71(2): 231–303
15. 15.
James I.N.(1995). Introduction to circulating atmospheres. Cambridge Atmospheric and Space Science Series. Cambridge University Press, LondonGoogle Scholar
16. 16.
Jin S., Xin Z.P.(1995). The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48(3): 235–276
17. 17.
Johnson R.H., Rickenbach T.M., Rutledge S.A., Ciesielski P.E., Schubert W.H.(1999). Trimodal characteristics of tropical convection. J. Atmos. Sci. 12, 2397–2418Google Scholar
18. 18.
Katsoulakis M.A., Tzavaras A.E. (1997). Contractive relaxation systems and the scalar multidimensional conservation law. Commun. Part. Diff. Eqns. 22(1-2): 195–233
19. 19.
Khouider B., Majda A.J.(2005). A non-oscillatory balanced scheme for an idealized tropical climate model: Part I: algorithm and validation. Theor. Comp. Fluid Dyn. 19(5): 331–354
20. 20.
Khouider B., Majda A.J.(2005). A non-oscillatory balanced scheme for an idealized tropical climate model: Part II: nonlinear coupling and moisture effects. Theor. Comp. Fluid Dyn. 19(5): 355–375
21. 21.
Khouider B., Majda A.J.(2006). A simple multicloud parameterization for convectively coupled tropical waves. Part I: linear analysis. J. Atmos. Sci. 63, 1308–1323
22. 22.
Khouider B., Majda A.J.: A simple multicloud parameterization for convectively coupled tropical waves. Part II: Nonlinear simulations. J. Atmos. Sci. (2006, in press)Google Scholar
23. 23.
Lax P.D.(1957). Hyperbolic systems of conservation laws. II. Commun. Pure Appl. Math. 10, 537–566
24. 24.
Madden R.A., Julian P.R.(1994). Observations of the 40–50-day tropical oscillation – a review. Mon. Wea. Rev. 122, 814–837
25. 25.
Majda A.(1981). A qualitative model for dynamic combustion. SIAM J. Appl. Math. 41(1): 70–93
26. 26.
Majda A.(1984). Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences.vol. 53. Springer, Berlin Heidelberg New YorkGoogle Scholar
27. 27.
Majda A. (2003). Introduction to PDEs and waves for the atmosphere and ocean Courant Lecture vol. 9. Notes in Mathematics. New York University Courant Institute of Mathematical Sciences, New YorkGoogle Scholar
28. 28.
Majda A.J., Khouider B., Kiladis G.N., Straub K.H., Shefter M.G.(2004). A model for convectively coupled tropical waves: nonlinearity, rotation, and comparison with observations. J. Atmos. Sci. 61, 2188–2205
29. 29.
Majda A.J., Shefter M.G.(2001). Models of stratiform instability and convectively coupled waves. J. Atmos. Sci. 58, 1567–1584
30. 30.
Majda A.J., Shefter M.G.(2001). Waves and instabilities for model tropical convective parameterizations. J. Atmos. Sci. 58, 896–914
31. 31.
Moncrieff M.W., Klinker E.(1997). Organized convective systems in the tropical western pacific as a process in general circulation models: a toga coare case-study. Q. J. R. Meteor. Soc. 123(540): 805–827
32. 32.
Nakazawa T.(1988). Tropical super clusters within intraseasonal variations over the western pacific. J. Meteor. Soc. Jpn. 66(6): 823–839Google Scholar
33. 33.
Neelin J.D., Held I.M.(1987). Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev. 115, 3–12
34. 34.
Neelin J.D., Held I.M., Cook K.H.(1987). Evaporation–wind feedback and low-frequency variability in the tropical atmosphere. J. Atmos. Sci. 44, 2341–2348
35. 35.
Neelin J.D., Yu J.Y.(1994). Modes of tropical variability under convective adjustment and the madden–julian oscillation. Part 1: analytical theory. J. Atmos. Sci. 51, 1876–1894
36. 36.
Neelin J.D., Zeng N.(2000). A quasi-equilibrium tropical circulation model—formulation. J. Atmos. Sci. 57, 1741–1766
37. 37.
Pauluis O., Majda A.J., Frierson D.M.W.: Propagation, reflection, and transmission of precipitation fronts in the tropical atmosphere. (in preparation) (2006)Google Scholar
38. 38.
Philander S.G.(1989). El Niño, La Niña, and the Southern Oscillation. International Geophysics Series. Academic, New YorkGoogle Scholar
39. 39.
Slingo J.M., et al.(1996). Intraseasonal oscillations in 15 atmospheric general circulation models: results from an amip diagnostic subproject. Clim. Dyn. 12(5): 325–357
40. 40.
Smith, R.K. (eds): The physics and parameterization of moist atmospheric convection. NATO Science Series C: Mathematical and Physical Sciences. Kluwer, Dordrecht, (2004)Google Scholar
41. 41.
Straub K.H., Kiladis G.N.(2002). Observations of a convectively coupled kelvin wave in the eastern pacific itcz. J. Atmos. Sci. 59, 30–53
42. 42.
Wheeler M., Kiladis G.N.(1999). Convectively coupled equatorial waves: analysis of clouds and temperature in the wavenumber–frequency domain. J. Atmos. Sci. 56(3): 374–399
43. 43.
Wheeler M., Kiladis G.N., Webster P.J.(2000). Large-scale dynamical fields associated with convectively coupled equatorial waves. J. Atmos. Sci. 57(5): 613–640
44. 44.
Williams F.A.(1985). Combustion theory. Addison Wesley, Reading
45. 45.
Yano J.I., Emanuel K.A.(1991). An improved model of the equatorial troposphere and its coupling to the stratosphere. J. Atmos. Sci. 48, 377–389
46. 46.
Yano J.I., McWilliams J.C., Moncrieff M.W., Emanuel K.A.(1995). Hierarchical tropical cloud systems in an analog shallow-water model. J. Atmos. Sci. 52, 1723–1742

## Copyright information

© Springer-Verlag 2006

## Authors and Affiliations

1. 1.Department of Mathematics and Center for Atmosphere Ocean Science, Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA