Article Type:
Research Article

The Importance of the Horizontal Distribution of Heating during Extratropical Cyclone Development

Phillip J. Smith Department of Earth and Atmospheric Sciences, Purdue University, West Lafayette, Indiana

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Print Publication:
01 Oct 2000
DOI:
https://doi.org/10.1175/1520-0493(2000)128<3692:TIOTHD>2.0.CO;2
Page(s):
3692–3694
Full access

Abstract

Diagnostic and modeling results reveal that atmospheric heating typically acts to intensify extratropical cyclones. In addition, both the Petterssen–Sutcliffe and Zwack–Okossi development equations reveal that this relationship depends on the proportionality that exists between surface geostrophic vorticity tendency and the negative of the horizontal Laplacian of atmospheric heating. Because of this Laplacian relationship, the impact of a heating field with a given magnitude and vertical distribution depends on its horizontal distribution. This paper will show how horizontal heating distributions that differ by relatively small amounts over their entire extent can yield vorticity tendency responses that could contribute to either development or decay of an underlying cyclone.

Corresponding author address: Prof. Phillip J. Smith, Department of Earth and Atmospheric Sciences, Purdue University, 1397 Civil Engineering Building, West Lafayette, IN 47907-1397.

Abstract

Diagnostic and modeling results reveal that atmospheric heating typically acts to intensify extratropical cyclones. In addition, both the Petterssen–Sutcliffe and Zwack–Okossi development equations reveal that this relationship depends on the proportionality that exists between surface geostrophic vorticity tendency and the negative of the horizontal Laplacian of atmospheric heating. Because of this Laplacian relationship, the impact of a heating field with a given magnitude and vertical distribution depends on its horizontal distribution. This paper will show how horizontal heating distributions that differ by relatively small amounts over their entire extent can yield vorticity tendency responses that could contribute to either development or decay of an underlying cyclone.

Corresponding author address: Prof. Phillip J. Smith, Department of Earth and Atmospheric Sciences, Purdue University, 1397 Civil Engineering Building, West Lafayette, IN 47907-1397.

1. Introduction

The intensification of extratropical cyclones by atmospheric heating (either diabatic heating or temperature advection) has been demonstrated in many model studies (see, e.g., Pauley and Smith 1988; Kuo and Reed 1988; Reed et al. 1988; Manobianco 1989; Kuo and Low-Nam 1990; Uccellini 1990; Kuo et al. 1991) and diagnostic studies (see, e.g., Tracton 1973; Gyakum 1983a,b; Davis and Emanuel 1988; Roebber 1989; Hirschberg and Fritsch 1991a,b; Jusem and Atlas 1991;Lupo et al. 1992; Rolfson and Smith 1996).

Clearly, the extent to which cyclones respond to such heating depends on the magnitude of the heating itself. Further, it has been shown, again with both model and diagnostic analyses (Anthes and Keyser 1979; Hirschberg and Fritsch 1991a,b; Steenburgh and Holton 1993;Rausch and Smith 1996), that the response to heating depends on its vertical distribution.

A question that to the author’s knowledge has received little attention to date is the sensitivity of the surface response to the horizontal heating distribution. The objective of this note is to explore this latter aspect of the heating–cyclogenesis relationship. The methodology used is to represent the horizontal heating distribution as a set of simple analytic functions and then to examine the sensitivity of the surface geostrophic vorticity tendency to these functions.

2. Background discussion

The work of Sutcliffe (1939, 1947) and Petterssen (1955) and the more recent extension of these ideas by Zwack and Okossi (1986) reveal that surface development can be effectively diagnosed by examining the surface geostrophic vorticity tendency (∂ξgs/∂t). Further, the forcing of this tendency by a temperature changing process is proportional to the vertical integral of the Laplacian (∇2) of that process, that is,
i1520-0493-128-10-3692-e1
where TC is a process that is capable of producing a local temperature change, such as diabatic heating or horizontal temperature advection. The presence of ∇2 in (1) reveals that the ∂ξgs/∂t response depends on the horizontal TC distribution. Further, in order for ∂ξgs/∂t to be nonzero, the TC distribution must be nonlinear.

3. Analytic diagnosis

Since the TC field must be distributed in a nonlinear fashion, the following question arises: How sensitive is the surface response to the precise heating distribution? To answer this question three nonlinear analytic functions (represented in one dimension to simplify the problem) have been formulated to simulate three heating distributions (see Fig. 1). The first is the upper half of a sine wave with wavelength Lx,
i1520-0493-128-10-3692-e2
The second is a parabola defined on the same interval,
i1520-0493-128-10-3692-e3
The third is a full sine wave with wavelength Lx/2,
i1520-0493-128-10-3692-e4
The vertex coordinates and distance from vertex to focus required to specify (3) and the amplitude, leading constant, and phase lag quantities required to specify (4) were determined such that all Fn functions are defined with positive values over the same π/2 interval, maximized at the same point (x = Lx/4), and integrated to the same total heating (Lx/π). Multiplication of these functions by an appropriate dimensional constant would render the quantities in any heating unit and magnitude desired.
The corresponding Laplacians of the Fn functions [multiplied by −L2x to render them unitless and the sign given in (1)] are
i1520-0493-128-10-3692-e5
The Gn functions are illustrated in Fig. 2.

4. Results

Figure 2 shows that the inferred surface geostrophic vorticity tendency is quite sensitive to the horizontal heating distribution. The responses to F1 and F2 are positive (positive values of G1 and G2) throughout the domain. However, while G1 and G2 are of similar magnitude in the domain interior, they differ markedly in the exterior regions, despite the fact that F1 and F2 themselves are quite similar throughout. Comparing G1 and G2 with G3 reveals the most striking difference of all, the sign change that occurs in G3 at Lx/8 and 3Lx/8, thus rendering the response to F3 of opposite sign to that of F1 and F2 in the exterior regions.

Physically, these results can be explained by first realizing that in order to produce a surface geostrophic vorticity change, ∇2(TC), or Fn, must force horizontal divergence. In turn, since the second-order derivative of Fn is proportional to the horizontal divergence, the first derivative at any point on Fn (slope of Fn) must be proportional to the divergent component of the flow induced by Fn. For F1 and F2, the slope maximizes at the outer limits 0 and Lx/2 and decreases monotonically to zero at the midpoint Lx/4. Thus, although of different magnitude, both exhibit divergent flow throughout the domain. In contrast, for F3 the slope maximizes in the interior of the domain at the inflection points (Lx/8 and 3Lx/8) and then decreases monotonically to zero at both the midpoint and the outer limits. Thus, divergent (convergent) flow occurs inside (outside) the inflection points.

5. Discussion

In the context of cyclone development, the preceding results have important implications. They suggest that if the cyclone is located relatively near the heating maximum, one can expect the heating to intensify the cyclone regardless of the heating distribution, although the vigor of that development would be distribution sensitive. On the other hand, when the cyclone center is further removed from the heating maximum, both the vigor and the sign of the development would be distribution sensitive. In some cases, heating could actually weaken the cyclone. This result appears to be consistent with the results of Montgomery and Enagonio (1998), who found that, for a simulated tropical cyclone, convection placed near (far from) the cyclone center yielded significantly increased (slightly decreased) maximum mean tangential winds.

An examination of the composite cyclone diagnosis of Rolfson and Smith (1996) provides a measure of verification of a portion of these results. Examination of their Fig. 10 reveals moderate levels of latent heat release during each of the three cyclone development phases, and relatively small latent heating for weakening cyclone cases. The average geostrophic vorticity tendency response to these heating fields is given in Rolfson and Smith’s Fig. 7. In the three development phases, the heating yields the expected vorticity increase, while in the weakening phase the heating yields a vorticity decrease. Although the position of the heating maxima and the horizontal distributions of the heating are not known in these cases, these results confirm that positive heating fields can indeed result in cyclone weakening.

Finally, it is clear that the present results do not comment on the atmospheric response to more complex one-, two-, or three-dimensional TC distributions, nor on the many interactions and feedbacks that conspire to produce surface geostrophic vorticity changes. Even in these more complex scenarios, the Laplacian operator in (1) dictates a sensitivity to horizontal distribution that has heretofore received little attention in the scientific literature. The purpose of this paper has been to provide a simple demonstration of how profound that sensitivity can be.

Acknowledgments

This work was inspired by the M.S. thesis of Ms. Judy St-James and by discussions with her major professor, Dr. Peter Zwack, at the University of Québec at Montréal.

REFERENCES

  • Anthes, R. A., and D. Keyser, 1979: Test of a fine-mesh model over Europe and the United States. Mon. Wea. Rev.,107, 963–984.

  • Davis, C. A., and K. A. Emanuel, 1988: Observational evidence for the influence of surface heat fluxes on rapid maritime cyclogenesis. Mon. Wea. Rev.,116, 2649–2659.

  • Gyakum, J. R., 1983a: On the evolution of the QE II storm. Part I: Synoptic aspects. Mon. Wea. Rev.,111, 1137–1155.

  • ——, 1983b: On the evolution of the QE II storm. Part II: Dynamic and thermodynamic structure. Mon. Wea. Rev.,111, 1156–1173.

  • Hirschberg, P. A., and J. M. Fritsch, 1991a: Tropopause undulations and the development of extratropical cyclones. Part I: Overview and observations from a cyclone event. Mon. Wea. Rev.,119, 496–517.

  • ——, and ——, 1991b: Tropopause undulations and the development of extratropical cyclones. Part II: Diagnostic analysis and conceptual model. Mon. Wea. Rev.,119, 518–550.

  • Jusem, J. C., and R. Atlas, 1991: Diagnostic evaluation of numerical model simulations using the tendency equation. Mon. Wea. Rev.,119, 2936–2955.

  • Kuo, Y. H., and R. J. Reed, 1988: Numerical simulation of an explosively deepening cyclone in the eastern Pacific. Mon. Wea. Rev.,116, 2081–2105.

  • ——, and S. Low-Nam, 1990: Prediction of nine explosive cyclones over the western Atlantic Ocean with a regional model. Mon. Wea. Rev.,118, 3–25.

  • ——, R. J. Reed, and S. Low-Nam, 1991: Effects of surface energy fluxes during the early development and rapid intensification stages of seven explosive cyclones in the western Atlantic. Mon. Wea. Rev.,119, 457–476.

  • Lupo, A. R., P. J. Smith, and P. Zwack, 1992: A diagnosis of the explosive development of two extratropical cyclones. Mon. Wea. Rev.,120, 1490–1523.

  • Manobianco, J., 1989: Explosive east coast cyclogenesis: Numerical experimentation and model-based diagnostics. Mon. Wea. Rev.,117, 2384–2405.

  • Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci.,55, 3176–3207.

  • Pauley, P. M., and P. J. Smith, 1988: Direct and indirect effects of latent heat release on a synoptic-scale wave system. Mon. Wea. Rev.,116, 1209–1235.

  • Petterssen, S., 1955: A general survey of factors influencing development at sea level. J. Meteor.,12, 36–42.

  • Rausch, R. L. M., and P. J. Smith, 1996: A diagnosis of a model-simulated explosively developing extratropical cyclone. Mon. Wea. Rev.,124, 875–904.

  • Reed, R. J., A. J. Simmons, M. D. Albright, and P. Unden, 1988: The role of latent heat release in explosive cyclogenesis: Three examples based on ECMWF operational forecasts. Wea. Forecasting,3, 217–229.

  • Roebber, P. J., 1989: The role of surface heat and moisture fluxes associated with large-scale ocean current meanders in maritime cyclogenesis. Mon. Wea. Rev.,117, 1676–1694.

  • Rolfson, D. M., and P. J. Smith, 1996: A composite diagnosis of synoptic-scale extratropical cyclone development over the United States. Mon. Wea. Rev.,124, 1084–1099.

  • Steenburgh, W. J., and J. R. Holton, 1993: On the interpretation of geopotential height tendency equations. Mon. Wea. Rev.,121, 2642–2645.

  • Sutcliffe, R. C., 1939: Cyclonic and anticyclonic development. Quart. J. Roy. Meteor. Soc.,65, 518–524.

  • ——, 1947: A contribution to the problem of development. Quart. J. Roy. Meteor. Soc.,73, 370–383.

  • Tracton, M. S., 1973: The role of cumulus convection in the development of extratropical cyclones. Mon. Wea. Rev.,101, 573–592.

  • Uccellini, L. W., 1990: Processes contributing to the rapid development of extratropical cyclones. Extratropical Cyclones: The Erik Palmén Memorial Volume, G. W. Newton and E. O. Holopainen, Eds., Amer. Meteor. Soc., 81–105.

  • Zwack, P., and B. Okossi, 1986: A new method for solving the quasigeostrophic omega equation by incorporating surface pressure tendency data. Mon. Wea. Rev.,114, 655–666.

Fig. 1.
Fig. 1.

Analytic functions simulating three horizontal heating distributions

Citation: Monthly Weather Review 128, 10; 10.1175/1520-0493(2000)128<3692:TIOTHD>2.0.CO;2

Fig. 2.
Fig. 2.

Laplacians (multiplied by −L2x of the functions illustrated in Fig. 1

Citation: Monthly Weather Review 128, 10; 10.1175/1520-0493(2000)128<3692:TIOTHD>2.0.CO;2

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  • Anthes, R. A., and D. Keyser, 1979: Test of a fine-mesh model over Europe and the United States. Mon. Wea. Rev.,107, 963–984.

  • Davis, C. A., and K. A. Emanuel, 1988: Observational evidence for the influence of surface heat fluxes on rapid maritime cyclogenesis. Mon. Wea. Rev.,116, 2649–2659.

  • Gyakum, J. R., 1983a: On the evolution of the QE II storm. Part I: Synoptic aspects. Mon. Wea. Rev.,111, 1137–1155.

  • ——, 1983b: On the evolution of the QE II storm. Part II: Dynamic and thermodynamic structure. Mon. Wea. Rev.,111, 1156–1173.

  • Hirschberg, P. A., and J. M. Fritsch, 1991a: Tropopause undulations and the development of extratropical cyclones. Part I: Overview and observations from a cyclone event. Mon. Wea. Rev.,119, 496–517.

  • ——, and ——, 1991b: Tropopause undulations and the development of extratropical cyclones. Part II: Diagnostic analysis and conceptual model. Mon. Wea. Rev.,119, 518–550.

  • Jusem, J. C., and R. Atlas, 1991: Diagnostic evaluation of numerical model simulations using the tendency equation. Mon. Wea. Rev.,119, 2936–2955.

  • Kuo, Y. H., and R. J. Reed, 1988: Numerical simulation of an explosively deepening cyclone in the eastern Pacific. Mon. Wea. Rev.,116, 2081–2105.

  • ——, and S. Low-Nam, 1990: Prediction of nine explosive cyclones over the western Atlantic Ocean with a regional model. Mon. Wea. Rev.,118, 3–25.

  • ——, R. J. Reed, and S. Low-Nam, 1991: Effects of surface energy fluxes during the early development and rapid intensification stages of seven explosive cyclones in the western Atlantic. Mon. Wea. Rev.,119, 457–476.

  • Lupo, A. R., P. J. Smith, and P. Zwack, 1992: A diagnosis of the explosive development of two extratropical cyclones. Mon. Wea. Rev.,120, 1490–1523.

  • Manobianco, J., 1989: Explosive east coast cyclogenesis: Numerical experimentation and model-based diagnostics. Mon. Wea. Rev.,117, 2384–2405.

  • Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci.,55, 3176–3207.

  • Pauley, P. M., and P. J. Smith, 1988: Direct and indirect effects of latent heat release on a synoptic-scale wave system. Mon. Wea. Rev.,116, 1209–1235.

  • Petterssen, S., 1955: A general survey of factors influencing development at sea level. J. Meteor.,12, 36–42.

  • Rausch, R. L. M., and P. J. Smith, 1996: A diagnosis of a model-simulated explosively developing extratropical cyclone. Mon. Wea. Rev.,124, 875–904.

  • Reed, R. J., A. J. Simmons, M. D. Albright, and P. Unden, 1988: The role of latent heat release in explosive cyclogenesis: Three examples based on ECMWF operational forecasts. Wea. Forecasting,3, 217–229.

  • Roebber, P. J., 1989: The role of surface heat and moisture fluxes associated with large-scale ocean current meanders in maritime cyclogenesis. Mon. Wea. Rev.,117, 1676–1694.

  • Rolfson, D. M., and P. J. Smith, 1996: A composite diagnosis of synoptic-scale extratropical cyclone development over the United States. Mon. Wea. Rev.,124, 1084–1099.

  • Steenburgh, W. J., and J. R. Holton, 1993: On the interpretation of geopotential height tendency equations. Mon. Wea. Rev.,121, 2642–2645.

  • Sutcliffe, R. C., 1939: Cyclonic and anticyclonic development. Quart. J. Roy. Meteor. Soc.,65, 518–524.

  • ——, 1947: A contribution to the problem of development. Quart. J. Roy. Meteor. Soc.,73, 370–383.

  • Tracton, M. S., 1973: The role of cumulus convection in the development of extratropical cyclones. Mon. Wea. Rev.,101, 573–592.

  • Uccellini, L. W., 1990: Processes contributing to the rapid development of extratropical cyclones. Extratropical Cyclones: The Erik Palmén Memorial Volume, G. W. Newton and E. O. Holopainen, Eds., Amer. Meteor. Soc., 81–105.

  • Zwack, P., and B. Okossi, 1986: A new method for solving the quasigeostrophic omega equation by incorporating surface pressure tendency data. Mon. Wea. Rev.,114, 655–666.

Fig. 1.

Analytic functions simulating three horizontal heating distributions
Fig. 2.

Laplacians (multiplied by −L2x of the functions illustrated in Fig. 1
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