The Effects of Latent Heating on Cold Frontal Speeds and Accelerations from a Potential Vorticity Perspective

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Abstract

REEVES, HEATHER DAWN. The Effects of Latent Heating on Cold Frontal Speeds and Accelerations from a Potential Vorticity Perspective. (Under the direction of Gary M. Lackmann.)

The effects of latent heating on frontal speed are investigated. It is conjectured that the existence of prefrontal latent heating leads to faster translation speeds and that the development of latent heating in the prefrontal zone can lead to frontal acceleration. A case study of a cold front where the attending precipitation band propagated into the prefrontal zone is presented. This front accelerated at the same time the precipitation moved into the prefrontal zone. Through inspection of the potential vorticity tendencies due only to latent heating, there is evidence that latent heating did alter the wind flow in the prefrontal zone, which may have contributed to positive frontogenetic tendencies in the prefrontal zone.

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THE EFFECTS OF LATENT HEATING ON COLD

FRONTAL SPEEDS AND ACCELERATIONS FROM A

POTENTIAL VORTICITY PERSPECTIVE

by

HEATHER DAWN REEVES

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the

requirements of the Degree of Master of Science

MARINE, EARTH, AND ATMOSPHERIC SCIENCES

Raleigh, North Carolina 2002

APPROVED BY: Yuh-Lang Lin Roscoe R. Braham

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Dedication

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Biography

Heather Dawn Reeves was born on December 2, 1973 in Gratiot Co. Michigan. She graduated from Mt. Pleasant High School in 1992 and from Central Michigan

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Acknowledgments

I would like to thank the members of my advisory committee, Gary Lackmann, Roscoe Braham, and Yuh-Lang Lin. This research was funded by the National Science Foundation (Grant # ATM-0079425). Data used in the research was provided by Unidata. Appreciation is extended to NCAR/MMM for providing the MM5 model. Thank you also the other members of the Weather and Forecasting group at North Carolina State University, especially Mike Brennan for his assistance with MM5 and Margaret Puryear and Scott Kennedy for their assistance. For your assistance in other matters, thanks to Tim Arnold and Bob Derr. I would also like to acknowledge the love

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List of Figures

2.1 The effects of confluence (a) and diffluence (b) on the quasi-horizontal potential temperature (dashed lines) gradient on an isobaric surface. Con-fluence (difCon-fluence) increases (decreases) the temperature gradient (after Bluestein 1986). . . 23 2.2 The effects of tilting on the vertical potential temperature (dashed line)

gradient: (a) frontogenesis; (b) frontolysis (after Bluestein 1986). . . 24 2.3 Illustration of the adjustment process that occurs when a horizontal

tem-perature gradient is increased through the action of geostrophic deforma-tion: Isobars (solid lines), forces (solid vectors) and induced ageostrophic circulation (dashed vectors) (a) Initial time; pressure gradient force (PGF) and Coriolis force are in balance. (b) Later time; the temperature gradient has increased, pressure gradient force is greater than Coriolis force. This incites the ageostrophic circulation (dashed vectors) (after Bluestein 1986). 25 2.4 (a) Illustration of geostrophic stretching increasing the potential

tempera-ture (dashed lines) gradient due to the geostrophic wind (vectors), where (∂vg/∂y)(∂θ/∂y)>0; (b) Illustration of geostrophic shearing turning the

potential temperature gradient parallel to thex-axis, where (∂θ/∂x)(∂ug/∂y)>

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2.5 Streamlines depicting the ageostrophic circulation for positive forcing at y = p = 0 for the case where static stability dominates over absolute vorticity (a) and for the case where absolute vorticity dominates over static stability (b). (after Eliassen 1962). . . 26 2.6 A surface of constant m in they₋pplane assuming constant geostrophic

absolute vorticity and a constant quasihorizontal temperature gradient (after Bluestein 1993). . . 27 2.7 (a) Vertical profile of a latent heating maximum (grey contours, maximum

denoted by an “L”) superposed on a north-south oriented cold front (isen-tropes - dashed). Pressure levels (black, solid countours) are vertically displaced due to latent heating and the wind field undergoes geostrophic adjustment(shown by J

- wind flowing out of the page, and N

- wind flowing into the page) . (b) Plan view of cold front and the wind field (black vectors) after geostrophic adjustment has 0 (isentropes - dashed). 27 2.8 Illustration of a mature front. The cloudy region represents typical cloud

configurations for this time in the front’s life as well as the location of latent heating; isobars - solid contours, isentropes - dashed contours, a

vertical profile; wind into the page - N

, wind out of the page J

, b plan view, vectors represent wind pattern . . . 28 2.9 Illustration of a katafront. The cloudy region represents typical cloud

configurations for this type of front; isobars solid contours, isentropes -dashed contours, a vertical profile; wind into the page - N

,b plan view, vectors represent wind pattern . . . 29

3.1 The potential vorticity associated with a standard atmosphere (Ps) and

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3.2 Illustration of a condensational heating and evaporative cooling couplet, such as would be expected in the region of a stationary, precipitating cloud. The isentropes (dashed contours) are closer together between the heated and cooled regions. Therefore, there exists a PV anomaly between the two regions. . . 43 3.3 Idealized illustration of PV redistribution due to shear. (a) Wind profile,

increasing height, in the y-direction only. (b) x, z-cross section showing latent heating (solid contours) with a maximum (L), and isentropes (dotted lines). Areas with positive (negative) PV tendencies are given. . . 43 3.4 Vertical cross section of a cold front. The cold air is given by the shaded

region. The symbol J

represents wind flowing into the page and N rep-resents wind flowing out of the page. The larger the symbol, the stronger the wind flow that is represented. The position of the latent heating maxi-mum, the PV maximaxi-mum, and the region of highest, positive PV tendencies are given in the figure. . . 44 3.5 Horizontal cross section of a cold front showing isentropes (dashed lines),

a region of positive PV tendencies (shaded - see legend in figure), and the wind flow associated with the developing PV anomaly (vectors). . . 45

4.1 Manual surface analyses of equivalent potential temperature (red, dashed - contoured every 5 K) and stations reporting precipitation (light shading - moderate precipitation, dark shading - heavy precipitation) for 18 UTC 28 January 2001 - 21 UTC 29 January 2001. . . 65 4.1 continued . . . 66 4.2 Infrared satellite image of eastern United States for (a) 00 UTC 29 January

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4.3 Isochrones of the surface cold front from 00 UTC to 21 UTC 29 Jan-uary 2001 as derived from manual analyses of sea level pressure, potential temperature and winds. The values given along 32◦N are the distances traversed by the front between analysis times along 32◦N. . . 68 4.4 Graph showing frontal speed according to four different estimation

meth-ods (see legend in figure). . . 69 4.5 Plan view of regions with advecting winds greater than or equal to the

speed of the front (frontal speed is given in the lower left corner of each 1) - shaded, and wet-bulb potential temperature (thin, solid - contoured every 1◦C). . . 70 4.6 Radar reflectivity and wet-bulb potential temperature at 975 hPa (solid

black - contoured every 1◦C) for 00 UTC 29 - 21 UTC 29 January 2001 (missing 15 UTC data). . . 71 4.6 continued . . . 72 4.7 Radar reflectivity (shaded), parameterized latent heating rate at 900 hPa

(thick, solid - contoured every 10 K s−1 _{up to 50 K s}−1_{, then every 50 K}

s−1) and 975 hPa wet bulb potential temperature (thin, solid - contoured every 1◦C) for 00 UTC - 18 UTC 29 January 2001. . . 73 4.8 Horizontal cross section showing potential vorticity (shaded - the layer

shown is given in the lower left corner of each sub-figure, see legend in figure) and wet-bulb potential temperature (thin, solid - contoured every 2◦C). . . 74 4.9 Horizontal cross section showing frontogenesis (shaded, see legend) and 975

hPa wet-bulb potential temperature (thin, solid - contoured every 1◦C). 75 4.10 Vertical cross section showing latent heating rate (shaded, see legend),

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4.11 Horizontal cross section showing potential vorticity (shaded - see legend in Figure 4.8), 975 hPa wet-bulb potential temperature (thin, solid, con-toured every 1◦C), and 875 hPa HAPV tendencies (same contour interval as LHPV in Figure 4.10) . . . 77 4.12 Horizontal cross section showing [0 vorticity (shaded - see legend in Figure

4.8), 975 hPa wet-bulb potential temperature (thin, solid, contoured every 1◦C), and 875 hPa LHPV tendencies (contoured as in Figure 4.10) . . . 78 5.1 Model domains used for numerical sensitivity tests . . . 99 5.2 Horizontal cross section showing model derived reflectivity (shaded - see

legend in figure) and 975 hPa wet-bulb potential temperature (contoured every 1◦C) for the control simulation. . . 100 5.2 continued. . . 101 5.3 Graph showing the speed of the front in the control simulation from 20

UTC 28 January 2001 to 21 UTC 29 January 2001. The darker curve shows the speed after a three-hour average and cubic splines interpolation was applied. The lighter curve represents the hourly average speeds plotted with a linear interpolation. . . 102 5.4 Horizontal cross section showing 975 hPa wet-bulb potential temperature

(contoured every 1◦C) and regions where the 950-900 hPa layer averaged advecting winds were moving faster than or equal to the speed at which the front was translating (shaded) for the control simulation. . . 103 5.5 continued. . . 104 5.6 Vertical cross section showing wet-bulb potential temperature (contoured

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5.7 Graph showing the speed of the front in the control simulation (blue) and the fake dry simulation (green) from 20 UTC 28 January 2001 to 21 UTC 29 January 2001. . . 107 5.8 Horizontal cross section showing model derived reflectivity (shaded - see

legend in figure) and 975 hPa wet-bulb potential temperature (contoured every 1◦C) for the fake dry simulation. . . 108 5.8 continued. . . 109 5.9 Vertical cross section showing wet-bulb potential temperature (contoured

every 1◦C) for the fake dry simulation. . . 110 5.9 continued. . . 111 5.10 Horizontal cross section showing hourly precipitation totals ending at the

time shown (shaded - see legend in figure), 925 hPa wind barbs (one full feather = 10 m s−1, on half feather = 5 m s−1) and isotachs (contoured every 10 m s−1_{in thick lines), and 975 hPa wet-bulb potential temperature}

(contoured every 1◦C in thin lines) for the control simulation. . . 112 5.10 continued. . . 113 5.11 Horizontal cross section showing hourly precipitation totals ending at the

(contoured every 1◦C in thin lines) for the fake dry simulation. . . 114 5.11 continued. . . 115 5.12 Graph showing the speed of the front in the control simulation (blue) and

the no cloud simulation (purple) from 20 UTC 28 January 2001 to 21 UTC 29 January 2001. . . 116 5.13 Vertical cross section showing wet-bulb potential temperature (contoured

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5.13 continued. . . 118 5.14 Graph showing the speed of the front in the control simulation (blue), the

0 simulation (red), and the nocps simulation (orange) from 20 UTC 28 January 2001 to 21 UTC 29 January 2001. . . 119 5.15 Horizontal cross section showing model derived reflectivity (shaded - see

legend in figure) and 975 hPa wet-bulb potential temperature (contoured every 1◦C) for the 0 simulation. . . 120 5.15 continued. . . 121 5.16 Horizontal cross section showing model derived reflectivity (shaded - see

legend in figure) and 975 hPa wet-bulb potential temperature (contoured every 1◦C) for the nocps simulation. . . 122 5.16 continued. . . 123 5.17 Vertical cross section showing wet-bulb potential temperature (contoured

every 1◦C) and LHPV tendencies (shaded - see legend in figure, the scale is 10−12 PVU s−1) for the 0 simulation. . . 124 5.17 continued. . . 125 5.18 Vertical cross section showing wet-bulb potential temperature (contoured

every 1◦C) and LHPV tendencies (shaded - see legend in figure, the scale is 10−12 PVU s−1) for the nocps simulation. . . 126 5.18 continued. . . 127 5.19 Horizontal cross section showing hourly precipitation totals ending at the

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5.20 Horizontal cross section showing hourly precipitation totals ending at the time shown (shaded - see legend in figure), 925 hPa wind barbs (one full feather = 10 m s−1, on half feather = 5 m s−1) and isotachs (contoured every 10 m s−1_{in thick lines), and 975 hPa wet-bulb potential temperature}

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Chapter 1 Introduction

1.1 Motivation

This research is an investigation into the effects latent heating has on the speed at which cold fronts move. More specifically, is the position of latent heating relative to the surface cold front related to the translation speed of the front?

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to a change in cold frontal speed has been investigated in this research.

Other research efforts have been made in understanding the relationship between frontal speed and precipitation. Charney and Fritsch (1999) and Bryan and Fritsch (2000) recently explored the problem of discrete propagation. As it is defined in the Bryan and Fritsch paper, discrete propagation is “non-continuous frontal motion” where the front has a “rapid” or “sudden” forward advancement “well ahead of the weather feature that has persisted for hours.” Bryan and Fritsch describe how discrete propagation occurs:

As the frontal system nears [prefrontal troughs], sometimes these troughs undergo frontogenesis and become more intense in temperature gradient and wind shift than the approaching cold front. Simultaneously, the original front undergoes frontolysis until it ceases to exist.

The prefrontal troughs referred to above are created by warm sector rainbands that prop-agate well into the prefrontal zone, completely disengaging from the frontal environment. Discrete propagation has been documented in other studies as well (Wilson and Stern 1985; Wilson et al. 1987; Hanstrum et al. 1990; Hutchinson and Bluestein 1998). How-ever, the literature reveals little information about what happens to a front’s speed as the warm sector rainband first moves into the prefrontal zone and is still in close proximity to the front.

1.2 Hypothesis

Latent heating, when it is located above the surface cold front, can act frontogenetically. Directly, it acts to increase temperature gradients via differential heating. Indirectly, it may also increase frontogenesis by increasing confluence and shearing across the frontal zone. Furthermore, the temperature gradient could be increased by differential temper-ature advection associated with the cyclonic circulation induced by the latent heating.

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can act frontogenetically across the frontal zone by increasing confluence and shearing, it can also increase confluence and shearing in the prefrontal zone. These processes may be strong enough to lead to a sensible increase in the near surface temperature gradient. Cold fronts are commonly analyzed as the leading edge of the surface temperature gradient. Hence, if prefrontal latent heating is acting to increase the temperature gradient in the prefrontal zone, the leading edge of the temperature gradient will be ahead of where it would be if there were no prefrontal latent heating (assuming all else is equal). Therefore, it seems logical to conclude that fronts with prefrontal latent heating should move faster than those without (all else being equal).

Consider again the precipitation bands defined by Parsons and Hobbs (1983). The warm sector rainband is described as initially developing above the surface cold front and propagating into the prefrontal zone. If prefrontal precipitation is associated with faster translation speeds, then it makes sense that the movement or development of precipitation into the prefrontal zone should lead to frontal acceleration. The hypothesis for this research is that latent heating, when located in the prefrontal zone of a cold front, leads to faster frontal translation speeds than if there were no prefrontal latent heating. This implies, then, that the movement of latent heating into the prefrontal zone can lead to frontal acceleration.

1.3 Relevance of Thesis

From these arguments, it follows that in order for the timing of frontal acceleration to be correctly forecasted, the position of precipitation with respect to the surface cold front needs to be correctly forecasted. Quantitative precipitation forecasting (QPF) has been documented as one of the weakest aspects of numerical weather forecasting. (Emanuel et al. 1995; Olson et al. 1995; Wang and Seaman 1997; Fritsch et al. 1998).

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heat-models typically do not completely resolve such precipitation. Therefore, the ability of a model to effectively handle convective precipitation is dependent on how well the con-vective parameterization scheme (CPS) operates in the given situation. It is sensible to conclude, then, that a model’s ability to capture the development and propagation of convective lines of precipitation is pivotal to the ability of a model to capture frontal acceleration. If the latent heating is underestimated, the model may fail to capture the frontal acceleration or have the timing of it too late. Conversely, if the latent heating is overestimated, the model may time the acceleration too early. On a similar note, if the NWP model fails to propagate a squall line into the prefrontal zone at the proper time, it may fail to capture frontal acceleration.

1.4 Research Objectives

The goals of this research are to

• establish whether the prefrontal latent heating is associated with faster translation speeds, and

• determine whether prefrontal latent heating can lead to frontal acceleration. This will be accomplished by examining a case study of an accelerating cold front and performing sensitivity experiments using the data from that event.

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1.5 References

Bryan, G.H. and J.M. Fritsch, 2000: Discrete propagation of surface fronts in a convective environment: Observations and theory. J. Atmos. Sci.,57, 2041-2060.

Charney, J.J. and J.M. Fritsch, 1999: Discrete frontal propagation in a nonconvective environ-ment. Mon. Wea. Rev.,127, 2083-2101.

Emanuel, K.A. and co-authors, 1995: Report of the first prospectus development team of the U.S. Weather Research Program to NOAA and the NSF.Bull. Amer. Meteor. Soc.,76, 1194-1208.

Fritsch, J.M. and co-authors, 1998: Quantitative precipitation forecasting: Report of the eighth prospectus development 1, U.S. Weather Research Program. Bull. Amer. Me-teor. Soc.,79, 285-299.

Hanstrum, B.N., K.J. Wilson, and S.L. Barrell, 1990: Prefrontal troughs over southern Aus-tralia. Part II: A case study of frontogenesis. Wea. Forecasting,5, 22-31.

Hutchinson, T.A. and H.B. Bluestein, 1998: Prefrontal wind-shift lines in the plains of the United States. Mon. Wea. Rev.,126, 141-166.

Parsons, D.B. and P.V. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VII: Formation, development, inter-action and dissipation of rainbands. J. Atmos. Sci.,40, 559-579.

Wang, W., and N.L. Seaman, 1997: A comparison study of convective parameterization schemes in a mesoscale model. Mon. Wea. Rev.,125, 252-278.

Wilson K.J. and H. Stern, 1985: The Australian summertime cool change. Part I: Synoptic and subsynoptic scale aspects. Mon. Wea. Rev.,113, 177-201.

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Chapter 2 Frontal Theory

As stated in Chapter 1, the hypothesis for this research is that latent heating, being a frontogenetic process, can lead to frontal acceleration should it develop in the prefrontal zone of a cold front. In this chapter, the theoretical arguments behind the hypothesis are presented. The fundamental equations that dictate motions in the frontal environment are presented in Section 2.1 Diabatic effects on frontogenesis and the ageostrophic circu-lation are discussed in Section 2.2. The hypothesis is reintroduced and fully explained in Section 2.3. A summary of the contents in this chapter is provided in Section 2.4.

2.1 Frontogenesis and the Ageostrophic Circulation

2.1.1 Frontogenesis in Two Dimensions

Petterssen (1936) introduced the frontogenesis function. The frontogenesis function is a measure of the time tendency of the quasi-horizontal temperature gradient following air parcel motion. Frontogenesis (F) is given by

F = D

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Substituting the thermodynamic equation, Dθ Dt = _p 0 p κ ₁ cp dQ

dt , (2.2)

into equation 2.1 gives the 2-dimensional frontogenesis function.

F = _∂v ∂y ∂θ ∂y p

| {z }

A + _∂ω ∂y_p ∂θ ∂p

| {z }

B − 1 cp _p 0 p κ _∂ ∂y p _dQ dt

| {z }

C

. (2.3)

Term A represents the effect of confluence or diffluence on the temperature gradient as illustrated by Figure 2.1. Qualitatively, it is easy to see how confluence acts to tighten the temperature gradient by pushing the isentropes closer together and diffluence acts to spread the isentropes apart; leading to frontogenesis in regions of confluence and frontolysis in regions of diffluence. Mathematically, the argument is equally simple. In Figure 2.1a, both∂v/∂yand∂θ/∂y are negative. Therefore,F is positive. In Figure 2.1b, ∂θ/∂y is still negative, but∂v/∂yis positive. Hence,F is negative. Therefore, confluence is a frontogenetic process and diffluence is a frontolytic process in this idealized example. In general, one must consider the orientation of the axis of dilatation to the isotherms in order to determine whether confluence is frontogenetic or not.

The second term in equation 2.3, term B, represents the how the tilting of isentropic surfaces effects frontogenesis. Figure 2.2 illustrates this effect. In Figure 2.2a, ∂ω_∂y < 0 and ∂θ_∂p < 0, which implies F > 0. For Figure 2.2b, ∂ω_∂y > 0 and ∂θ_∂p < 0 which means F < 0. A qualitative examination is equally valuable to the mathematical one. Notice that for the frontogenesis case, the distance between isentropes at the 0 time (∆y) is greater than the distance at the later time (∆y0). This means that tilting of the frontal interface into a more vertical orientation leads to a stronger temperature gradient along the surface: a frontogenetical process. Again, the angle the axis of dilatation makes with the isotherms must be considered in the general case.

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In making the assumption that all isentropes are parallel to thex-axis, a final term, the shearing term, was not included in equation 2.3. According to the more general equation for frontogenesis, including shearing, the dependence on whether shearing and confluence act frontogenetically or frontolytically is on the angle the axis of dilatation makes with the isentropes. Since inclusion of a discussion on the general frontogenesis equation does not add to further arguments, it will not be included here.

Equation 2.3 is the two dimensional form of the frontogenesis function. There is also a three dimensional form. Although there are additional terms, the forcing is still the same: confluence/diffluence, tilting, and diabatic heating. These forcings are referred to as geostrophic deformation. Since inclusion of the three dimensional form of the frontogenesis equation does not add to the arguments presented in future sections, it will not be included either.

In Chapter 1, the argument was made that by increasing the temperature gradient in the prefrontal zone, the cold front is moved forward. This claim is made because changes in the surface thermal gradient incite changes throughout other parts of the atmosphere.

2.1.2 Dynamical Effects of Frontogenesis

Namias and Clapp (1949) explained the dynamical effects an increase in the thermal gradient at the surface has on other parts of the atmosphere. These effects can be described mathematically using the thermal wind equation. Assuming

Dgθ

Dt = 0 , (2.4)

it can be shown (Hoskins et al. 1978) that ∂~vg

∂p =− R f0p

_p p0

κ

(~k_{× ∇}pθ) . (2.5)

This equation is called the thermal wind equation. It relates the vertical shear profile to the quasi-horizontal temperature gradient. If _|∇pθ| is increased, the vertical shear

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characterized by a relatively tight temperature gradient will have greater shear than those with relatively weaker gradients, all else being equal.

Consider Figure 2.3a which shows balanced flow across a layer bounded by the pressure surfacespandp+∆p. Suppose the temperature gradient across this layer is increased. An increase of the virtual temperature gradient affects the thickness of the layer (according to the hypsometric equation), which is reflected between the initial time and the later time (Figure 2.3b). The change in thickness leads to an increase in the pressure gradient force (PGF), which aloft, is directed toward cold air, and below, is directed toward warm air. The induced circulation due to the imbalance between pressure gradient force and Coriolis force is shown by the dashed lines in Figure 2.3b.

The direction of flow in the vertical branches of the circulation is dictated by the conti-nuity equation, which is given by (assuming no u-relative wind):

∂v ∂y =−

∂ω

∂p . (2.6)

According to equation 2.6, air rises in the warm sector and sinks in the cold sector. This circulation is called the ageostrophic circulation. Since rising air cools due to expansion, there is cooling in the warm sector. Similarly, there is warming due to compression in the cold sector. In this sense, the temperature gradient is partly relieved, and the atmosphere is closer to being in thermal wind balance.

Further restoration to the thermal wind balance is gained through rotational effects (assuming the ageostrophic circulation exists sufficiently long enough to be affected by the Earth’s rotation). The branch of the circulation aloft, moving north, will be deflected to the east. At lower levels, the flow will be deflected to the west. Therefore, ∂~vg/∂p

is increased. By reducing the temperature contrast via the ageostrophic circulation and increasing the vertical shear, the thermal wind balance is restored.

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2.1.3 The Sawyer-Eliassen Equation

Since strong temperature gradients usually exist along cold fronts, ageostrophic circu-lations generally also are present along cold fronts. It would make sense that if the ageostrophic circulation is strong, it could advect quantities, such as temperature and moisture. This could detectably alter the frontal environment. Quasi-geostrophic theory neglects ageostrophic advection, so it is not a suitable 0 to use to quantify cold frontal movements. The equation of motion (equation 2.7) and the thermodynamic equation (equation 2.8) according to the geostrophic momentum approximation are given below (Hoskins and Bretherton 1972).

∂~vg

∂t + (~vg+~va)· ∇p~vg +ω ∂~vg

∂p =−f(~k×~va) . (2.7) ∂θ

∂t + (~vg +~va)· ∇pθ+ω ∂θ ∂p = 1 cp _p p0 κ_dQ

dt . (2.8)

Assuming the isentropes are oriented parallel to thex-axis, the flow parallel to the front is straight (i.e. ∂vg

∂x = 0), and ua= 0, then equations 2.5, 2.6, 2.7 and 2.8 can be combined

into an equation that relates the ageostrophic circulation to geostrophic forcing; the Sawyer-Eliassen equation. The Sawyer-Eliassen equation is given by

∂2_ψ

∂y2

− ∂θ

∂p R f0p

_p p0

κ

| {z }

A

+ ∂

2_ψ

∂y∂p

2∂ug ∂p

| {z }

B

+∂

2_ψ

∂p2

f0 −

∂ug

∂y

| {z }

C

= 2 R f0p

_p p0 κ_∂θ ∂y ∂vg ∂y | {z }

D

+∂θ ∂x

∂ug

∂y | {z }

E

− R

cpf p

∂ ∂y

_dQ dt

| {z }

F

(2.9)

where ψ is the ageostrophic streamfunction, given by ∂2ψ

∂p2 = −

∂va

∂p (2.10)

∂2ψ ∂y2 =

∂ω

∂y (2.11)

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dia-When the right hand side of equation 2.9 is positive, there is rising air on the warm side and sinking air on the cold side of the front. Such a circulation is called thermally direct. Thermally indirect circulations happen when cold air rises and warm air sinks. The greater the forcing, the stronger the ageostrophic circulation.

Terms A and C dictate the ellipticity of the circulation. When static stability (∂θ/∂p) is much smaller in magnitude than absolute vorticity (f0−∂ug/∂y),

∂2_ψ

∂y2 >>

∂2_ψ

∂p2

⇒ ∂ω ∂y >>

∂va

∂p ,

and the vertical part of the circulation has a greater magnitude than the horizontal part. In the opposite sense, when static stability dominates over absolute vorticity, the horizontal branch of the circulation dominates (see Figure 2.5

The tilt of the circulation is dictated by the vertical shear, term C. The tilt of the ageostrophic circulation is parallel to the absolute momentum surfaces (m), where

m=ug−f0y . (2.12)

An absolute momentum surface is shown in Figure 2.6 for a location with constant geostrophic absolute vorticity and a constant quasihorizontal temperature gradient. The slope of the absolute momentum surface is unique for each point in space. It is given by

− _dp dy m =

f0−

∂ug

∂y

−∂ug ∂p

. (2.13)

Using the thermal wind relation, equation 2.13 becomes

− _dp dy m =

f0−

∂ug

∂y

− R f0p

∂T ∂y

. (2.14)

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Notice that in both the equation for frontogenesis (2.3) and the Sawyer-Eliassen equation (2.9) the direct effects of diabatic heating and cooling are accounted for. The interest in this research is on the effects condensational heating has on frontogenesis and the ageostrophic circulation in the prefrontal zone. However, it is beneficial to examine other sources of diabatic heating and cooling that are instrumental in changing the temperature gradient and the ageostrophic circulation.

2.2 Diabatic Processes

2.2.1 Differential Heating Due to Cloud Cover

Differential heating can result from differing cloud cover across the frontal zone. When there exist clouds on the cold side and clear skies on the warm side of a front, the blockage of insolation at the surface can increase the thermal gradient. In the opposite sense, cloud cover on the warm side of a cold front and clear skies on the other leads to frontolysis. Assuming a mildly wet surface, Gallus and Segal (1999) estimate that the daily increase of temperature (for average winter conditions) on the warm side of a frontal zone without cloud cover is between 2.5◦₋5◦C. This increase is relatively modest when compared to strong frontal gradients, so may be of only secondary importance. Koch et al. (1995) found in a numerical sensitivity test that the upward directed branch of the ageostrophic circulation was increased from 16 cm s−1 _{to 22 cm s}−1 _{when clouds were introduced on}

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2.2.2 Differential Heating Due to Differing Land Surface

Char-acteristics

Differential heating can also occur as the result of differing land surface characteristics such as soil moisture content. On the one hand, increased soil moisture can lead to a decreased thermal gradient when the ground on the warm side is moister because the specific heat of wet soil is higher than that for dry soil (assuming equal soil composition) (Segal et al. 1993; Koch et al. 1997). Other evidence suggests that increased soil moisture can enhance thermodynamic forcing on the warm side of the thermal gradient (De Ridder 1997). Overall, Koch et al. (1997) found that the sensitivity of thermal gradients to differing soil moisture characteristics was not large for their simulated cold front (1◦ - 2◦ of the 0 thermal gradient (22◦)). Gallus and Segal (1999) state that there may be an “optimal surface wetness that maximizes both available soil moisture for cloud 0 and the low-level convergence that enables the development of a favorable convective environment,” and that further sensitivity tests need to be performed to 0 the role surface wetness plays in precipitation potential.

2.2.3 Evaporative Cooling

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lation at low levels from 6 m s−1 to 7 m s−1, but also increased the along-the-front flow ahead of the surface cold front (the low-level jet) from 35 m s−1 _{to 40 m s}−1_{. This study}

is not unique: other studies indicate even light precipitation has detectable influences on the thermal gradient (Oliver and Holzworth 1953; Gallus and Johnson 1995; Gallus and Segal 1999).

2.2.4 Condensational Heating

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differ-ences in confluence and shearing between the two simulations directly to latent heating: stating “frontal circulations can be affected greatly by latent heating... These processes are important not only for their direct effects, but also for their interactions with other dynamical processes which influence the circulation such as deformation.” So, by in-creasing the surface thermal gradient, condensational heating can act to intensify surface cold fronts. An increase in the surface temperature gradient leads to a more robust ageostrophic circulation. However, this is not the only way the ageostrophic circulation is affected by condensational heating.

First, condensational heating reduces the rate of cooling experienced by rising air parcels. Therefore, the ageostrophic circulation will be less effective at reducing the temperature gradient. So, in order to restore the thermal wind balance, the ageostrophic circulation will need to be increased. Second, assuming a lifted air parcel is condensationally heated enough that its temperature is greater than the surrounding environment, it will no longer require a lifting mechanism in order to rise. It will rise due to its own buoyancy. Thus, the ageostrophic circulation is increased. This is the finding of Ross and Orlanski (1982). They compared a simulated cold front without condensational heating to one with. The inclusion of condensational heating resulted in a stronger ageostrophic circulation and more convection than the dry simulation.

To summarize, shearing, confluence and the thermal gradient are increased under regions where condensational heating is occurring. Furthermore, condensational heating leads to an increase in rising motions. Hence, condensational heating, when positioned above the surface frontal zone, can act to increase the temperature gradient across the frontal zone.

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2.3 Frontal Movement

2.3.1 Advection by Synoptic Scale

Previous research has shown that frontal motion is dictated by the phase speed of the synoptic scale disturbance. This is the finding of Ryan and Wilson (1985). However, their definition of a cold front differed from the one used herein, in that they defined the cold front as the “final change line” or the final transition zone within the larger frontal zone itself. This would place the cold front toward the rear of the frontal zone, rather than at the leading edge.

2.3.2 Gravity Wave Theory

Explanations of frontal movement after “frontal collapse” frequently rely on gravity cur-rent theory, where the motion of the front is dictated by a large density gradient across the frontal zone. Usually, this large density gradient is attributed to evaporative cool-ing behind the frontal zone. There have been many observational studies where the researchers likened frontal motion to that of a gravity current (Carbone 1982; Hobbs and Persson 1982; Bond and Fleagle 1985; Coulman et al. 1985; Seitter and Muench 1985; Shapiro et al. 1985).

As Smith and Reeder (1988) point out, there are some problems with using gravity current theory to explain frontal motions. The equation for the speed of the current is determined using representative values of the potential temperature on either side of the front, as well as representative values for the depth of the cold air, the pressure rise, the speed of the tail wind and the Froude number. Smith and Reeder question how these representative values are determined, since these quantities frequently vary along the length of cold fronts. They go on to state that the addition of precipitation into the system complicates the issue, since outflow boundaries often have the characteristics of both gravity currents and cold fronts and could be mistakenly identified.

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There are many cases where gravity current theory has been correctly applied (Bond and Fleagle 1985; Koch and Kocin 1991). However, there are authors, besides Smith and Reeder who have examined the problem and found that gravity current theory does not apply in all cases. Both Garner (1989) and Thorpe and Clough (1991) found that although gravity waves were generated at the time of frontal collapse, they did not significantly alter the pattern of frontogenesis. Yet as Garrett (1988) noted, some fronts do move faster than the normal component of the wind at any level behind the front, so some process, other than advection, must be involved in determining the speed of the front. In cases where gravity current theory does not apply, what process is responsible for those cases where cold fronts propagate faster than the advecting winds?

2.3.3 Hypothesis

What happens when precipitation develops in the prefrontal zone of cold fronts? As was discussed in Section 2.2, prefrontal precipitation acts to reduce contrasts across the surface frontal zone. Based on this, it is tempting to assume the front should slow down. However, the hypothesis for this research is that the front will accelerate. Since latent heating is frontogenetic, it makes sense that the development of latent heating in the prefrontal zone should be accompanied by the development of some front-like characteristics in the prefrontal zone. Figure 2.9 shows a schematic of this process. Notice that while front-like contrasts are generated in the prefrontal zone, the contrasts across the original baroclinic zone associated with the cold front are being reduced. What this figure illustrates is, in essence, the argument of the hypothesis.

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advected. Hence, the front will appear to propagate through the environment.

Observational evidence from previous research does seem to correlate well with the hy-pothesis. Chen and Lin (1999) examined a cold front before frontal collapse and found that the main process acting to translate the front was advection. Sanders (1999) noted a cold front that after collapse, had a reduced temperature contrast and moved faster. Also after collapse, Sanders noted the convergence zone stayed continually ahead of the temperature contrast. Finally, he observed that the front moved faster than the advecting wind after collapse.

The above arguments, in fact, are in keeping with an early observational study per-formed by Sansom (1951) who noted that cold fronts generally start life as anafronts and transition to katafronts as the parent cyclone matures. Anafronts are well character-ized by Figure 2.8, with post-frontal precipitation and a strongly tilted frontal interface. Katafronts, on the other hand are better represented by Figure 2.9. The more distinctive traits of this frontal type are prefrontal precipitation with little or no post-frontal pre-cipitation and a nearly vertical frontal interface. Most of the previous work on anafronts and katafronts has revealed that, as a general rule, katafronts move faster than anafronts (Bergeron 1937; Petterssen 1940; Godske et al. 1957; Browning 1990). From this, it would make sense that fronts transitioning from an anafront structure to a katafront structure should undergo some sort of acceleration.

2.4 Summary

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Particular attention was given to how condensational heating can act to increase the surface thermal gradient and the ageostrophic circulation. As Parsons and Hobbs (1983) explain, warm sector rainbands are convective lines that propagate relative to the frontal zone into the warm sector of the parent cyclone. It is hypothesized that the development of condensational heating in the prefrontal zone can lead to frontal 0.

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2.5 References

Baldwin, D., E.Y. Hsie, and R.A. Anthes, 1984: Diagnostic studies of a two-dimensional simu-lation of frontogenesis in a moist atmosphere. J. Atmos. Sci.,41, 2686-2700.

Bergeron, T., 1937: On the physics of fronts. Bull. Amer. Meteor. Soc.,18, 265-275.

Bluestein, H.B., 1986: Atmospheric fronts: An observational perspective. Chap 9, Mesoscale Meteorology and Forecasting(P. Ray ed.), Amer. Meteor. Soc., Boston, 173-215.

—–, 1993: Synoptic-Dynamic Meteorology in Midlatitudes: Volume II. Oxford University Press, New York 594pp.

Bond, N.A. and R.G. Fleagle, 1985: Structure of a cold front over the ocean. Quart. J. Roy. Meteor. Soc,111, 739-759.

Browning, K.A., 1990: Organization of clouds and precipitation in extratropical cyclones. Ex-tratropical Cyclones: The Erik Palm´en Memorial Volume, Amer. Meteor. Soc., 129-153.

Carbone, R.E., 1982: A severe winter squall line. Stormwide hydrodynamic structure. J. At-mos. Sci.. 39, 258-279.

Chen, G.T.J. and K.C. Lin, 1999: A diagnostic case study of a winter low-level front over southern China. Mon. Wea. Rev.,127, 1096-1107.

Coulman, C.E., J.R. Colquhoun, R.K. Smith and K. McInnes, 1985: Orographically forced cold fronts-mean structure and motion. Bound. Layer. Meteor.,32, 57-83.

Cunningham, R.M. and F. Sanders, 1987: Into the teeth of the gale: The remarkable advance of a cold front at Grand Manon. Mon. Wea. Rev.,115, 2450-2462.

De Ridder, K., 1998: Land surface processes and the potential for convective precipitation. J. Geophys. Res,102, 30 085-30 090.

Eliassen, A., 1962: On the vertical 0 in frontal zones. Geofys. Publ.,24, 147-160.

Gallus, W.A., and R.H. Johnson, 1995: The dynamics of circulations within the stratiform re-gions of squall lines. Part I: The 11 June PRE-STORM system. J. Atmos. Sci., 52, 2161-2187.

—–, and J.R. Bresch, 1997: An intense small-scale wintertime vortex in the midwest United States. Mon. Wea. Rev.,125, 2787-2807.

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Garner, S.T., 1989: Fully Lagrangian numerical solutions of unbalanced frontogenesis and frontal collapse. J. Atmos. Sci.,46, 717-739.

Garrett, J.R., 1988: Summertime cold fronts in southeast Australia - behavior and low-level structure of main frontal types. Mon. Wea. Rev.,116, 636-649.

Godske, C.L., T. Bergeron, J. Bjerknes and R.C. Bundgaard, 1957: Dynamic meteorology and weather forecasting. Amer. Meteor. Soc. and Carnegie Institution of Washington, 800pp.

Hobbs P.V., and P.O.G. Persson, 1982: The mesoscale and microscale structure and organiza-tion of clouds and precipitaorganiza-tion in midlatitude cyclones. Part V: The substructure of narrow cold-frontal rainbands. J. Atmos. Sci.. 39, 280-295.

Hoskins B.J., and F.P. Bretherton, 1972: Atmospheric frontogenesis models: Mathematical formulation and solution. J. Atmos. Sci.,29, 11-37.

—–, I. Draghici, and H.C. Davies, 1978: A new look at theω-equation. Quart. J. Roy. Meteor. Soc. ,104, 31-38.

Huang, H.C., and K.A. Emanuel, 1991: The effects of evaporation on frontal circulations. J. Atmos. Sci.,48, 619-628.

Koch, S.E., and P.J. Kocin, 1991: Frontal contraction processes leading to the formation of an intense narrow rainband. Meteor. Atmos. Phys.,46, 123-154.

—–, J.R. McQueen, and V.M. Karyampudi, 1995: A numerical study of the effects of differen-tial cloud cover on cold frontal structure and dynamicsJ. Atmos. Sci.,52, 937-964.

—–, A. Aksakal, and J.T. McQueen, 1997: The influence of mesoscale humidity and evapotran-spiration fields on a model forecast of cold frontal squall line Mon. Wea. Rev., 125, 384-409.

Namias, J., and P.F. Clapp, 1949: Confluence theory of the high tropospheric jet stream. J. Meteor.,6, 330-336.

Oliver, V.J., and G.C. Holzworth, 1953: Some effects of the evaporation of widespread precip-itation on the production of fronts and on changes in frontal slopes and motion. Mon. Wea. Rev.,81, 141-151.

Petterssen, S., 1936: A contribution to the theory of frontogenesis. Geofys. Publ.,11, 1-27. —–, 1940: Weather Analysis and Forecasting. McGraw-Hill Company Inc., New York and

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simulation. J. Atmos. Sci.,39, 296-327.

Sanders, F., 1999: A short-lived cold front in the southwest United States. Mon. Wea. Rev.,

127, 2395-2403.

Sansom, H.W., 1951: A study of cold fronts over the British Isles. Quart. J. Roy. Meteor. Soc.,77, 96-120.

Sawyer, J.S., 1956: The vertical circulation at meteorological fronts and its relation to fronto-genesis. Proc. Roy. Soc. London., A234, 346-362.

Segal, M., W. L. Physick, J.E. Heim, and R.W. Arritt, 1993: The enhancement of cold front temperature contrast by differential cloud cover. Mon. Wea. Rev.,121, 867-873.

Seitter, K.L., and H.S. Muench, 1985: Observation of a cold front with rope cloud. Mon. Wea. Rev.,113, 840-848.

Shapiro, M.A., T. Hampel, D. Rotzoll and F. Mosher, 1985: The frontal hydraulic head: A microscale (1 km) triggering mechanism for mesoconvective weather systems. Mon. Wea. Rev.,113, 1166-1183.

Smith, R.K. and M.J. Reeder, 1988: On the movement and low-level structure of cold fronts.

Mon. Wea. Rev.,116, 1927-1944.

Thorpe, A.J. and S.A. Clough, 1991: Mesoscale dynamics of cold fronts: Structures described by drop-soundings in FRONTS 87. Quart. J. Roy. Meteor. Soc.,117, 903-941.

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2.6 Figures

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Figure 2.4: (a) Illustration of geostrophic stretching increasing the potential temperature (dashed lines) gradient due to the geostrophic wind (vectors), where (∂vg/∂y)(∂θ/∂y)>

0; (b) Illustration of geostrophic shearing turning the potential temperature gradient parallel to the x-axis, where (∂θ/∂x)(∂ug/∂y)>0 (b) (after Bluestein 1986).

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Figure 2.6: A surface of constant m in the y₋p plane assuming constant geostrophic absolute vorticity and a constant quasihorizontal temperature gradient (after Bluestein 1993).

Figure 2.7: (a) Vertical profile of a latent heating maximum (grey contours, maximum denoted by an “L”) superposed on a north-south oriented cold front (isentropes - dashed). Pressure levels (black, solid countours) are vertically displaced due to latent heating and the wind field undergoes geostrophic adjustment(shown by J

- wind flowing out of the page, and N

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Figure 2.8: Illustration of a mature front. The cloudy region represents typical cloud configurations for this time in the front’s life as well as the location of latent heating; isobars - solid contours, isentropes - dashed contours, a vertical profile; wind into the page - N

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Figure 2.9: Illustration of a katafront. The cloudy region represents typical cloud con-figurations for this type of front; isobars - solid contours, isentropes - dashed contours,

a vertical profile; wind into the page - N

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Chapter 3 Potential Vorticity

3.1 Why use PV?

At the end of Chapter 2, the question was posed, how can the effects of latent heating on the low-level winds be quantified? Potential vorticity (PV) seems to be useful for this research in that it is a product of both latent heating and vertical vorticity, thus it can be used to assess how latent heating perturbs the dynamics of the surrounding environment. Persson (1995) stated, most elegantly, why PV is such a useful diagnostic:

...both synoptic-scale and mesoscale variations of the PV field can be dynam-ically significant. These variations can simultaneously indicate that a certain process may have happened, such as diabatic heating or slantwise convec-tion, and imply that another process may occur in the future, such as ‘flow induction’ or symmetric instability.

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provides a summary of the important points of this chapter and a discussion of how PV can be meaningfully used in this research.

3.2 Potential Vorticity Review

3.2.1 Barotropic Potential Vorticity

In 1939, Rossby noted that many synoptic scale flow features could be well modeled by assuming absolute vorticity is conserved for barotropic, two-dimensional, frictionless, adiabatic, horizontal motion.

ζa

h = constant (3.1)

(wherehis the fluid depth) following the fluid column. Examination ofζa/hprovides one

with a simple method of examining those processes that modulate vorticity, in particular stretching and advection. Increasing the fluid depth, h, means that absolute vorticity must also be increased. This implies that the air parcel must either move to a higher latitude or increase its relative vorticity.

In 1940, Rossby generalized his original equation (equation 3.1). For barotropic, friction-less, adiabatic, hydrostatic flow of a finite depth to

f+ζθ

−δp/g = constant , (3.2)

following an air parcel. The meaning of equation 3.2 is essentially the same as equation 3.1. However, the vertical distance in equation 3.2 is a function of pressure rather than a specified vertical distance. Therefore, as the mass per unit area of a fluid parcel is increased, either the latitude must be increased or the relative vorticity (on an isentropic surface) must be increased.

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there is a potential for creating vorticity by changing the latitude of an air parcel or by adiabatically changing the vertical distance between isentropes within the air parcel. Equation 3.2 is conserved in any layer of fluid where the horizontal wind is non-divergent. Mid-tropospheric, synoptic scale flows are usually non-divergent, thus barotropic PV provides a good diagnostic for these flows.

3.2.2 Ertel’s Potential Vorticity

Another version of PV for non-hydrostatic, adiabatic flow was developed by Ertel (1942).

P V =₋g∂θ

∂pζaθ . (3.3)

This equation is more generalized than Rossby’s because it is valid for baroclinic flow. Since cold fronts are baroclinic zones, equation 3.3 is better suited for this research. Using values for typical, synoptic scale flow,

P V ∼= 10−6m2s−1Kkg−1

≡ 1P V U (3.4)

(Hoskins et al. 1985).

3.2.3 Definition of PV Anomaly

Throughout this writing, the term “PV anomaly” is used extensively. In order to be precise with this terminology, a specific 0 of the term PV anomaly has been made. Consider the standard, or background potential vorticity which is given by

Ps =−f g

∂θ

∂p (3.5)

(Hoskins et al. 1985). Figure 3.1 shows Ps profiles assuming a standard atmosphere for

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increases with increasing altitude only slightly beneath the tropopause. Typical low-level, mid-latitude PV values hover around 0.25 PVU.

According to Cooper et al. (1992), a typical diabatic heating rate of 8 K d−1 and an absolute vorticity of 2f, leads to the generation of a positive PV anomaly at low levels at an approximate rate of 0.5 PVU d−1. Taking this value into consideration and that of the background PV, a positive PV anomaly, for the purposes of this research is defined as any region where PV is greater than 0.5 PVU.

3.2.4 Potential Vorticity as a Diagnostic

Starr and Neiburger (1940) recognized that PV could be used to diagnose diabatic pro-cesses since it is conserved for adiabatic flow. They speculated that PV anomalies should exist in regions of latent heating or other significant diabatic processes. According to equation 3.3, PV should be higher in regions where the isentropes are more closely packed. Regions of latent heating/cooling should be accompanied by PV anomalies because latent heating and cooling both affect the isentropic field. Figure 3.2 shows a condensational heating and evaporative cooling couplet, such as might be expected for a precipitating cloud. In this figure, the isentropes between the regions of heating and cooling are closer together than those away from the heating and cooling couplet. Therefore, elevated values of PV should be found in this region. The PV anomaly is shown as the shaded area in Figure 3.2.

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3.3 Potential Vorticity and Cold Fronts

Thorpe and Emanuel (1985) were among the earliest in the new generation of “PV thinkers” to recognize that low-level PV is valuable for cold frontal research, citing that equivalent PV can be used to diagnose symmetric instabilities: a line of thinking that was later pursued by Joly and Thorpe (1990). Thorpe and Clough (1991) provided a list of why PV is useful for the specific purpose of studying cold fronts.

1. The distribution of PV, along with the potential temperature distribution on the boundaries, can be used to obtain the balanced flow from the invertibility principle. The invertibility principle is described in detail by Hoskins et al. (1985). The value of performing a PV inversion is that it allows for one to determine the exact contribution changes in the thermodynamic profile make to the wind field. Potential vorticity inversions for diagnosing winds in cyclone events have been the target of many research efforts (Davis and Emanuel 1991; Davis 1992; Stoelinga 1996; Martin and Marsili 2002) Not until recently has research focused on diagnosing the contribution cold frontal latent heating makes on the surrounding environment and the front itself via PV inversions. Among those efforts that have used a PV inversion with respect to cold frontal processes are Morgan (1999), who used a PV inversion to diagnose the contributions latent heating made to frontogenesis for a cold front event. He determined that latent heating can be significant, but that the contributions vary as the system in question matures. Based on a PV inversion of a cyclone event, Korner and Martin (2000) suggest that unbalanced flow near fronts can be an important mechanism acting to increase frontogenesis. Lackmann (2002) found that cold frontal latent heating contributed up to 40% of the speed in the low-level jet.

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advection or the dominance of latent heat release (see Figure 3.2).

Other work focusing on cold frontal PV anomalies involved establishing a connection between PV and positive feedback mechanisms in the vicinity of cold fronts or squall lines (Chan and Cho 1989; Raymond and Jiang 1990; Hertenstein and Schubert 1991). According to these authors, the positive feedback mechanism occurs when a PV anomaly is embedded in a region of vertical shear and tilted isentropes. The cyclonic circulation associated with the PV anomaly, then, indicates that there are rising motions on the side of the anomaly where the flow must travel up isentropic surfaces. As this flow travels upward, the lifting condensation level may be reached and latent heating will be initiated on that side of the anomaly. This new region of latent heating will push isentropes closer together beneath itself and a new PV anomaly is born and the process can begin again. Some conclusions have also been drawn on the structure of PV anomalies generated by latent heating. The typical PV anomaly that is caused by cold frontal latent heating is described by both Chan and Cho (1989) and Joly and Thorpe (1990) as broad or wide. This notion follows from Bluestein (1993) who stated that the circulations associated with latent heating are of a larger scale than the latent heating anomaly itself.

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3.4 Potential Vorticity Redistribution

3.4.1 Creation of PV Anomalies

One of the primary attributes of PV is that it is conserved for adiabatic frictionless motion, so it can be used as an air parcel tracer. Haynes and McIntyre (1987) recognized that if PV is a tracer, the PV of a given air parcel is related to and can be calculated from the Lagrangian history of the diabatic and external forces acting on the parcel. This implies that the processes acting to increase or decrease the PV in an isentropic layer can be deduced without referring to the PV in adjacent isentropic layers. Therefore

1. there can be no net transport of Rossby-Ertel potential vorticity across any isentropic surface, and

2. potential vorticity can neither be created nor destroyed, within a layer bounded by two isentropic surfaces.

According to statement 1, fluxes of PV cannot travel across isentropic surfaces. Potential vorticity is confined to the layer in which it was generated. Statement 2 indicates that PV is not a quantity that can be introduced into a layer that does not intersect a boundary, such as the earth’s surface. Potential vorticity, can, however, be concentrated at certain places along an isentropic layer and it can be advected along the layer. Those processes that are responsible for the positioning and intensity of PV anomalies, according to Haynes and McIntyre are advection, friction, and latent heating/cooling.

3.4.2 Mathematics of Potential Vorticity Redistribution

The temporal changes in PV due to changes in latent heating can be expressed by: dP V

dt =−gζaθ ∂θ˙ ∂p | {z }

A

+g∂θ ∂p~k·(∇

˙ θ_× ∂~v

∂θ)

| {z }

B

. (3.6)

Considering only term A gives

dP V

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According to this term, PV is redistributed beneath latent heating rate maxima. There-fore, those regions beneath the greatest increases in latent heating should be experiencing the greatest increases in PV. In terms of cold frontal dynamics, as new regions of latent heating are being generated ahead of the current maximum, new PV anomalies are also being generated ahead of the current PV maximum. With time, the region of increasing latent heating will become the new latent heating maximum. Likewise, the region of strongest positive PV tendencies will become the new PV maximum. This theoretical argument is supported by the research of Schubert et al. (1989). They found that the positive PV anomalies generated by a squall line in a region of zero vertical wind shear were located just beneath the latent heat rate maximum.

Term B relates changed in potential vorticity to the shear profile: dP V

dt ∝g ∂θ ∂p~k·(∇

˙ θ_× ∂~v

∂θ) (3.8)

which can be rewritten as: dP V

dt ∝g ∂θ ∂p _∂v ∂θ ∂θ˙ ∂x − ∂u ∂θ ∂θ˙ ∂y . (3.9)

Figure 3.3 helps to illustrate the influence of shear on PV tendency. This figure shows the wind profile on the left side and, on the right side, a west-to-east profile of latent heating (solid contours, the maximum is denoted as ˙θmax) and isentropes (dotted lines

with θ1 < θ2, etc). In this situation, there is no u-component to the wind. Therefore,

equation 3.9 can be written as dP V

dt ∝g ∂θ ∂p

∂v ∂θ

∂θ˙

∂x . (3.10)

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3.4.3 PV Redistribution and Cold Frontal Propagation

Positive PV tendencies are an indication that the wind in that region is adopting cyclonic curvature and beginning to converge. These are both frontogenetic processes. If the contribution latent heating makes to the total PV tendency can be deduced, then whether or not latent heating is acting frontogenetically in the prefrontal zone can be determined. Because PV is redistributed downwind of latent heating rate maxima in a sheared envi-ronment, it is possible that latent heating could act to modify the winds in the prefrontal zone even if there is no prefrontal latent heating. Consider Figure 3.4 which shows a vertical profile of a cold front with a latent heating maximum located above the frontal zone. In this case, the greatest increases in PV due to latent heating are in the prefrontal zone. This implies that a cyclonic circulation is developing in the prefrontal zone. This developing cyclonic circulation opposes the cyclonic circulation across the frontal zone. Figure 3.5 shows a horizontal cross section of the winds associated with the PV anomaly and the developing PV anomaly across the frontal zone. Notice that while contrasts are developing in the prefrontal zone, the contrasts across the frontal zone are reduced. Therefore, latent heating is acting frontogenetically in the prefrontal zone and the leading edge of the temperature gradient will be ahead of where it would have been had there been no shear.

3.5 Summary

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3.6 References

Bluestein, H.B., 1993: Synoptic-Dynamic Meteorology in Midlatitudes: Volume II. Oxford Uni-versity Press, New York 594pp.

Browning, K.A. and T.W. Harrold, 1970: Air motion and precipitation growth at a cold front.

Quart. J. Roy. Meteor. Soc.,96, 369-389.

Cammas, J.-P., D. Keyser, G.M. Lackmann, and J. Molinari, 1994: Diabatic redistribution of potential vorticity 0 the development of an outflow jet within a strong extratropical cy-clone. Proc. Int. Symp. On the Life Cycles of Extratropical Cyclones, Vol II, Bergen, Norway, Geophysical Institute, University of Bergen. 403-409.

Chan, D.T., and H.R. Cho, 1989: Meso-β potential vorticity anomalies and rainbands: Part I: Adiabatic dynamics of potential vorticity anomalies. J. Atmos. Sci.,46, 1713-1723.

Cooper, I.M., A.J. Thorpe, and C.H. Bishop, 1992: The role of diffusive effects on potential vorticity in fronts. Quart. J. Roy. Meteor. Soc.,118, 629-647.

Davis, C.A., 1992: A potential-vorticity 0 of the importance of initial structure and condensa-tional heating in observed extratropical cyclogenesis. Mon. Wea. Rev.,120, 2409-2428.

—–, and K.A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev.,

119, 1929-1953.

—–, M.T. Stoelinga, and Y.H. Kuo, 1993: The integrated effect of condensation in numerical simulations of extratropical cyclogenesis. Mon. Wea. Rev.,121, 2309-2330.

Haynes, P.H., and M.E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci., 44, 828-841.

Hertenstein, R.F.A. and W.H. Schubert, 1991: Potential vorticity anomalies associated with squall lines. Mon. Wea. Rev.,119, 1663-1672.

Hoskins, B.J., M.E. McIntyre, and A.W. Robertson, 1985: On the use and significance of isen-tropic potential vorticity maps. Quart. Roy. Meteor. Soc.,111, 877-944.

Joly, A. and A.J. Thorpe, 1990: Frontal instability generated by tropospheric potential vortic-ity. Quart. J. Roy. Meteor. Soc.,116, 525-560.

Korner, S.O. and J.E. Martin, 2000: Piecewise frontogenesis from a potential vorticity perspec-tive: Methodology and a case study. Mon. Wea. Rev.,128, 1266-1288.

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transport in extratropical cyclones. Mon. Wea. Rev.,130, 59-74.

Martin, J.E. and N. Marsili, 2002: Surface cyclolysis in the north Pacific Ocean. Part II: Piece-wise potential vorticity diagnosis of a rapid cyclolysis event. Mon. Wea. Rev., 130, 1264-1281.

Morgan, M.C., 1999: Using piecewise potential vorticity inversion to diagnose frontogenesis. Part I: A partitioning of the Q-vector applied to diagnosing surface frontogenesis and vertical motion. Mon. Wea. Rev.,127, 2796-2821.

Persson, P.O.G, 1995: Simulations of the potential vorticity structure and budget of FRONTS 87 IOP8. Quart. J. Roy. Meteor. Soc.,121, 1041-1081.

Raymond, D.J. 1992: Nonlinear balance and PV thinking at large Rossby number. Quart. J. Roy. Meteor. Soc.,121, 1041-1081.

—– and H. Jiang, 1990: A theory for long-lived mesoscale convective systems. J. Atmos. Sci.,

47, 3067-3077.

Rossby, C.-G., 1939: Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J. Marine. Res.,2(1), 28-55.

—–, 1940: Planetary flow patterns in the atmosphere. Quart. J. Roy. Meteor. Soc.,66suppl. 58-87.

Schubert, W.H., S.R. Fulton, and R.A. Hertenstein, 1989: Balanced atmospheric response to squall lines. J. Atmos. Sci.,46, 2478-2483.

Stoelinga, M.T., 1996: A potential vorticity-based study of the role of diabatic heating and friction in a numerically simulated baroclinic cyclone. Mon. Wea. Rev.,124, 849-874.

Thorpe, A.J. and K.A. Emanuel, 1985: Frontogenesis in the presence of small stability to slant-wise convection. J. Atmos. Sci.,42, 1809-1824.

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3.7 Figures

Figure 3.1: The potential vorticity associated with a standard atmosphere (Ps) and zero

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Figure 3.2: Illustration of a condensational heating and evaporative cooling couplet, such as would be expected in the region of a stationary, precipitating cloud. The isentropes (dashed contours) are closer together between the heated and cooled regions. Therefore, there exists a PV anomaly between the two regions.

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Figure 3.4: Vertical cross section of a cold front. The cold air is given by the shaded region. The symbol J

represents wind flowing into the page and N

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The Effects of Latent Heating on Cold Frontal Speeds and Accelerations from a Potential Vorticity Perspective

THE EFFECTS OF LATENT HEATING ON COLD

FRONTAL SPEEDS AND ACCELERATIONS FROM A

POTENTIAL VORTICITY PERSPECTIVE

Contents

List of Figures

Chapter 1

Introduction

1.1

Motivation

1.2

Hypothesis

1.3

Relevance of Thesis

1.4

Research Objectives

1.5

References

Chapter 2

Frontal Theory

2.1

Frontogenesis and the Ageostrophic Circulation

2.1.1

Frontogenesis in Two Dimensions

2.1.2

Dynamical Effects of Frontogenesis

2.1.3

The Sawyer-Eliassen Equation

2.2

Diabatic Processes

2.2.1

Differential Heating Due to Cloud Cover

2.2.2

Differential Heating Due to Differing Land Surface

Char-acteristics

2.2.3

Evaporative Cooling

2.2.4

Condensational Heating

2.3

Frontal Movement

2.3.1

Advection by Synoptic Scale

2.3.2

Gravity Wave Theory

2.3.3

Hypothesis

2.4

Summary

2.5

References

2.6

Figures

Chapter 3

Potential Vorticity

3.1

Why use PV?

3.2

Potential Vorticity Review

3.2.1

Barotropic Potential Vorticity

3.2.2

Ertel’s Potential Vorticity

3.2.3

Definition of PV Anomaly

3.2.4

Potential Vorticity as a Diagnostic

3.3

Potential Vorticity and Cold Fronts

3.4

Potential Vorticity Redistribution

3.4.1

Creation of PV Anomalies

3.4.2

Mathematics of Potential Vorticity Redistribution

3.4.3

PV Redistribution and Cold Frontal Propagation

3.5

Summary

3.6