Mesoscale Meteorology in Midlatitudes

Paul Markowski and Yvette Richardson,

Penn State University, University Park, PA, USA

A John Wiley & Sons, Ltd., Publication

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Mesoscale Meteorology in Midlatitudes

Mesoscale Meteorology in Midlatitudes

Paul Markowski and Yvette Richardson,

Penn State University, University Park, PA, USA

A John Wiley & Sons, Ltd., Publication

This edition first published 2010, 2010 by John Wiley & Sons, Ltd

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We dedicate this book to our familiesMarisa, Nolan, & Shane

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Contents

Series Foreward xiPreface xiiiAcknowledgments xvList of Symbols xvii

PART I General Principles 11 What is the Mesoscale? 3

1.1 Space and time scales 31.2 Dynamical distinctions between the mesoscale

and synoptic scale 5

2 Basic Equations and Tools 112.1 Thermodynamics 112.2 Mass conservation 162.3 Momentum equations 172.4 Vorticity and circulation 212.5 Pressure perturbations 252.6 Thermodynamic diagrams 322.7 Hodographs 34

3 Mesoscale Instabilities 413.1 Static instability 413.2 Centrifugal instability 483.3 Inertial instability 493.4 Symmetric instability 533.5 Shear instability 58

PART II Lower Tropospheric Mesoscale Phenomena 714 The Boundary Layer 73

4.1 The nature of turbulent fluxes 734.2 Surface energy budget 824.3 Structure and evolution of the boundary layer 834.4 Boundary layer convection 88

viii CONTENTS

4.5 Lake-effect convection 934.6 Urban boundary layers 1034.7 The nocturnal low-level wind maximum 105

5 Air Mass Boundaries 1155.1 Synoptic fronts 1175.2 Drylines 1325.3 Outflow boundaries 1405.4 Mesoscale boundaries originating from differential

surface heating 149

6 Mesoscale Gravity Waves 1616.1 Basic wave conventions 1616.2 Internal gravity wave dynamics 1656.3 Wave reflection 1706.4 Critical levels 1726.5 Structure and environments of ducted mesoscale

gravity waves 1736.6 Bores 175

PART III Deep Moist Convection 1817 Convection Initiation 183

7.1 Requisites for convection initiation and the roleof larger scales 183

7.2 Mesoscale complexities of convection initiation 1897.3 Moisture convergence 1957.4 Elevated convection 197

8 Organization of Isolated Convection 2018.1 Role of vertical wind shear 2018.2 Single-cell convection 2068.3 Multicellular convection 2098.4 Supercellular convection 213

9 Mesoscale Convective Systems 2459.1 General characteristics 2459.2 Squall line structure 2499.3 Squall line maintenance 2539.4 Rear inflow and bow echoes 2609.5 Mesoscale convective complexes 265

CONTENTS ix

10 Hazards Associated with Deep Moist Convection 27310.1 Tornadoes 27310.2 Nontornadic, damaging straight-line winds 29210.3 Hailstorms 30610.4 Flash floods 309

PART IV Orographic Mesoscale Phenomena 31511 Thermally Forced Winds in Mountainous Terrain 317

11.1 Slope winds 31711.2 Valley winds 320

12 Mountain Waves and Downslope Windstorms 32712.1 Internal gravity waves forced by two-dimensional terrain 32712.2 Gravity waves forced by isolated peaks 33212.3 Downslope windstorms 33312.4 Rotors 342

13 Blocking of the Wind by Terrain 34313.1 Factors that govern whether air flows over or around

a terrain obstacle 34313.2 Orographically trapped cold-air surges 34613.3 Lee vortices 35113.4 Gap flows 358

PART V Appendix 367A Radar and Its Applications 369

A.1 Radar basics 369A.2 Doppler radar principles 371A.3 Applications 374

References 389

Index 399

Series Foreword

Advances in Weather and Climate

Meteorology is a rapidly moving science. New develop-ments in weather forecasting, climate science and observ-ing techniques are happening all the time, as shown bythe wealth of papers published in the various meteo-rological journals. Often these developments take manyyears to make it into academic textbooks, by which timethe science itself has moved on. At the same time, theunderpinning principles of atmospheric science are wellunderstood but could be brought up to date in the lightof the ever increasing volume of new and exciting obser-vations and the underlying patterns of climate change thatmay affect so many aspects of weather and the climatesystem.

In this series, the Royal Meteorological Society, in con-junction with Wiley–Blackwell, is aiming to bring togetherboth the underpinning principles and new developments

in the science into a unified set of books suitable forundergraduate and postgraduate study as well as being auseful resource for the professional meteorologist or Earthsystem scientist. New developments in weather and climatesciences will be described together with a comprehensivesurvey of the underpinning principles, thoroughly updatedfor the 21st century. The series will build into a com-prehensive teaching resource for the growing number ofcourses in weather and climate science at undergraduateand postgraduate level.

Series Editors

Peter Inness, University of Reading, UK

William Beasley,University of Oklahoma, USA

Preface

This text originated from course notes used in the under-graduate mesoscale meteorology class at PennsylvaniaState University. We assume that students have alreadyhad courses in atmospheric dynamics, thermodynamics,and synoptic meteorology. A mesoscale meteorology text-book likely will always be a ‘‘work in progress’’, giventhat so much of what we teach is constantly evolvingas observing and numerical modeling capabilities contin-ually improve. Another obvious challenge in preparinga reference on mesoscale meteorology is that the spe-cialty is extraordinarily broad, and in a way a catch-allfor essentially all atmospheric phenomena that are notdominated at one extreme by quasigeostrophic dynamicsor at the other extreme by the effects of small-scale tur-bulence. Thus, it is perhaps impossible to write a trulycomprehensive mesoscale meteorology textbook that canadequately address all of the mesoscale processes thatinfluence the weather in every corner of the world inimportant ways.

Our focus is midlatitude mesoscale phenomena. Thethermodynamics and dynamics of tropical convective clus-ters and hurricanes are therefore not included, nor is acomprehensive treatment of polar lows. It is our experiencethat these topics tend to be covered in tropical meteorol-ogy and synoptic meteorology courses, respectively, ratherthan in mesoscale meteorology courses. Other perhaps sur-prising omissions include jet streaks and lee cyclogenesis,and the treatment of fronts and frontogenesis might beconsidered by some to be rather abridged. Again, in ourexperience these topics also tend to be covered in courses onsynoptic meteorology. We also did not include chapters onupslope precipitation events or mesoscale modeling. Themost interesting aspects of the former topic are probably themicrophysical aspects (e.g. the seeder-feeder process) ratherthan the mesoscale dynamical aspects. Regarding mesoscalemodeling, even though numerous figures throughout thetext are derived from numerical simulations, we felt thatthis topic deserves an entire course by itself. It is possi-ble that we might reconsider including these topics in anexpanded future edition. We also caution the reader thatthe subject of atmospheric convection, particularly deep,

moist convection, is what drew us to meteorology in thefirst place and its study is what puts food on our tables. Itwill be obvious to the casual reader that this bias has notbeen well concealed.

The book is divided into four parts. Part I, GeneralPrinciples, begins by defining what is meant by the termmesoscale (Chapter 1). This requires the introduction ofsome basic dynamical concepts, such as the Rossby number,hydrostatic approximation, and pressure perturbations. InChapter 2 we present a more detailed review of the tools thatwill be needed for the rest of the book. Some readers maywish to skip Chapter 2. It might seem somewhat awkward tointroduce some dynamics in Chapter 1 and then review thebasic governing equations more thoroughly in Chapter 2,but the alternative – forcing readers to trudge througha lengthy review chapter to open a book before gettingto a description of the types of phenomenon that are thefocus of the book – seemed even less attractive. One of theconcepts in Chapter 1 is that mesoscale phenomena canbe driven by a variety of instabilities, unlike synoptic-scalemotions, which are driven almost exclusively by baroclinicinstability, at least in midlatitudes. Chapter 3 discusses thesemesoscale instabilities.

The remaining chapters in the book (Parts II–IV)deal with mesoscale phenomena. The phenomena can beattributed to either instabilities, topographic forcing, or, inthe case of air mass boundaries such as fronts and dry-lines, frontogenesis. There no doubt are a number of waysto organize mesoscale meteorology topics, as is evidencedby the fact that we did things differently at least the firstfour times we taught the course at Penn State. In Part IIwe explore mesoscale phenomena that are confined prin-cipally to the lower troposphere, for example, boundarylayer convection, air mass boundaries (e.g. fronts, dry-lines, sea breezes, outflow boundaries), and ducted gravitywaves. Part III treats the subject of deep moist convec-tion, including its initiation, organization, and associatedhazards. Part IV contains mountain meteorology topics.The basic idea in Part IV is to treat each of the followingin a separate chapter, in this order: (i) the simplest case– no ambient flow and only heating/cooling of sloped

xiv PREFACE

terrain, which results in thermally forced mountain andvalley circulations; (ii) the case of wind flowing over atopographic barrier, which excites gravity waves and occa-sionally leads to severe, dynamically induced downslopewinds; (iii) phenomena resulting when winds that impingeon a topographic barrier experience significant blocking,such as cold-air damming, wake vortices, and gap winds.

We lament that each of Parts II–IV themselves could bethe basis for entire textbooks. The scope of each chapterpurposely has been limited somewhat to facilitate the exam-ination of a wide range of mesoscale topics within the course

of a typical semester. In part for this reason, a ‘‘furtherreading’’ list also appears at the end of each chapter, whichcontains supplemental references not specifically cited inthe bibliography. We speculate that these listings might bemost valuable to graduate students seeking to supplementthe contents herein with more advanced readings. Finally,a ‘‘crash course’’ on radar meteorology is provided in anappendix. Radars are arguably the most important instru-ment in the observation of mesoscale phenomena. After all,the term mesoscale originated in a review paper on radarmeteorology.

Acknowledgments

We are grateful for all of the discussions with our friends andcolleagues over the years: Mark Askelson, Peter Bannon,Howie Bluestein, Harold Brooks, George Bryan, DonBurgess, Fred Carr, John Clark, Bill Cotton, Bob Davies-Jones, Chuck Doswell, David Dowell, Kelvin Droegemeier,Dale Durran, Evgeni Fedorovich, Bill Frank, Mike Fritsch,Kathy Kanak, Petra Klein, Sukyoung Lee, Doug Lilly, MattParker, Erik Rasmussen, Dave Schultz, Alan Shapiro, NelsShirer, Todd Sikora, Dave Stensrud, Jerry Straka, Jeff Trapp,Hans Verlinde, Tammy Weckwerth, Morris Weisman, LouWicker, Josh Wurman, John Wyngaard, George Young, andConrad Ziegler. We are especially appreciative of those whoreviewed earlier versions of this book: George Bryan, JohnClark, Chuck Doswell, Dale Durran, Evgeni Fedorovich,Bart Geerts, Thomas Haiden, Jerry Harrington, Steve Koch,Dennis Lamb, Sukyoung Lee, Doug Lilly, Matt Parker, DaveSchultz, Russ Schumacher, Alan Shapiro, Nels Shirer, ToddSikora, Hans Verlinde, Dave Whiteman, Josh Wurman,John Wyngaard, George Young, and Fuqing Zhang.

We also thank those who provided us with their originalphotographs or figures (all photographs are copyrighted bythe those credited in the figure captions): Nolan Atkins,Peter Blottman, Harold Brooks, George Bryan, FernandoCaracena, Brian Colle, Chris Davis, Chuck Doswell, JimDoyle, Dale Durran, Charles Edwards, Roger Edwards,Craig Epifanio, Marisa Ferger, Brian Fiedler, Jeff Frame,Bart Geerts, Roberto Giudici, Joel Gratz, Vanda Grubisic,Jessica Higgs, Richard James, Dave Jorgensen, Pat Kennedy,Jim LaDue, Bruce Lee, Dave Lewellen, Amos Magliocco,Jim Marquis, Brooks Martner, Al Moller, Jerome Neufeld,Eric Nguyen, Matt Parker, Erik Rasmussen, Chuck Robert-son, Paul Robinson, Chris Rozoff, Thomas S{a}vert, DaveSchultz, Jim Steenburgh, Herb Stein, Jeff Trapp, RogerWakimoto, Nate Winstead, Josh Wurman, Ming Xue, andConrad Ziegler. A number of staff at Penn State helpedus acquire several archived datasets that were used toconstruct some of the figures within the book, in addi-tion to providing virtually ‘‘24/7’’ computer support: ChadBahrmann, Chuck Pavloski, Art Person, and Bill Syrett.We also are grateful for the support and patience of Wiley,especially Rachael Ballard and Robert Hambrook. Some of

the figures contain numerical model output generated bythe Advanced Regional Prediction System (ARPS), devel-oped by the Center for the Analysis and Prediction ofStorms at the University of Oklahoma, and the BryanCloud Model, developed by George Bryan. Much of theradar imagery appearing in figures was displayed using theSOLOII software from the National Center for AtmosphericResearch.

Paul MarkowskiYvette Richardson

Work on this book began in the spring of 2001 whenI began preparing to teach the undergraduate mesoscalemeteorology course at Penn State for the first time. Muchof the inspiration at that time came from reviewingGreg Forbes’ lecture notes from the class, which I tookfrom Dr. Forbes in 1995 as an undergraduate meteorol-ogy major at Penn State. Dr. Forbes’ influence on myearly development—through his formal classroom lec-tures, undergraduate honors thesis mentorship, and simplyshared interests in convective storms—cannot be over-stated. I also likely would not be where I am today if notfor the opportunity to spend the summer after my junioryear in Norman, Oklahoma, as a Research Experiences forUndergraduates (REU) student. My mentor there, DaveStensrud, is one of the reasons I decided to pursue aPh.D. Another important aspect of my REU experience inNorman was the opportunity to participate in the Verifica-tion of the Origins of Rotation in Tornadoes Experiment(VORTEX). My experience in the field forever sealed myfate to follow a career path to research. It was throughVORTEX that I met Jerry Straka and Erik Rasmussen, whoconvinced me to attend the University of Oklahoma andwho served as my advisors. They were superb advisors,and it’s hard to say what their biggest contribution was.It was either their trust in me to allow me to work soindependently right from the start, or it was their tirelessand selfless willingness to discuss pretty much any aspect ofmy research or theirs at virtually any hour of the day. I alsosingle-out Bob Davies-Jones, with whom I chased storms

xvi ACKNOWLEDGMENTS

for a number of years while doing field work as a part of mygraduate research. Hours upon hours of watching the skyand listening to Bob’s assessments, in addition to discussingdynamics problems, benefited me in immeasurable ways.Finally, I am forever grateful for the support of my wife(also a meteorologist) throughout the project.

Paul Markowski

My path to authoring this book was somewhat cir-cuitous. I majored in physics as an undergraduate at theUniversity of Wisconsin-River Falls. The professors I hadthere were incredible teachers and mentors, and I willalways be indebted to Drs. Shepherd, Larson, Paulson, andBlodgett for providing me with a solid foundation. Myjourney into meteorology began with the Summer Instituteon Atmospheric Science at NASA-Goddard Space FlightCenter between my junior and senior year. It was therethat my husband (also a physics major) and I both real-ized that atmospheric science was an extremely interestingapplication of our physics backgrounds, and it is where wemet Kelvin Droegemeier, who represented the Universityof Oklahoma graduate program with such enthusiasm wecould not help but go there! I am grateful to Fred Carrwho served as the thesis advisor for my masters degree and

did his best to teach a physics student to understand actualweather! For my Ph.D., I decided to study severe stormswith Kelvin, and I am ever grateful for his undying supportand encouragement. It was through him that I learned to bea numerical modeler, and his markups of my manuscriptstaught me the essence of scientific writing. I also will neverforget having the opportunity to sit at the feet of theoret-ical giants Douglas Lilly and Robert Davies-Jones, both ofwhom always were willing to discuss difficult concepts andpass along their incredible insight. As I was finishing myPh.D., the University of Oklahoma allowed me to get myfeet wet in teaching as a Visiting Assistant Professor, andthrough this I determined that was the career path for me.Following my Ph.D., I had the wonderful opportunity of apost-doc position with Joshua Wurman, who did his bestto help a numerical modeler become an observationalist,before landing at the Pennsylvania State University as anassistant professor. It has been an interesting path, and onemade possible through the support of family and all ofthe friendships developed along the way. In particular, thispath was possible because of my husband who started outas my study partner in my Freshman year of college andhas fully supported my endeavors ever since.

Yvette Richardson

List of Symbols

α specific volume, angle a parcel displacement

makes with respect to the horizontal, angle of

axis of dilatation with respect to the x axis,

inclination angle of sloping terrain

α0 constant reference specific volume

αd specific volume of dry air

β angle between v and dl, latitudinal variation ofCoriolis parameter, angle between isentropes

and the axis of dilatation, between-beam angle

γ environmental lapse rate

�d dry adiabatic lapse rate

�m moist adiabatic lapse rate

�p parcel lapse rate

�ps pseudoadiabatic lapse rate

�rm reversible moist adiabatic lapse rate

δ horizontal divergence, displacement of a

streamline

δc displacement of the dividing streamline

δ vertically averaged horizontal divergence

ε ratio of gas constants for dry air and water vapor,

dissipation

ζ vertical vorticity component

ζ mean (environmental) vertical vorticity

ζ′

vertical vorticity perturbation

η meridional vorticity component

〈η〉 cross-section-averaged meridional vorticityη mean (environmental) meridional vorticity

ξ zonal vorticity component

ξ mean (environmental) zonal vorticity

θ potential temperature, radar beam azimuth angle

θ mean (environmental) potential temperature

〈θ〉 layer-averaged environmental potentialtemperature

θ a mean potential temperature at anemometer level

θ′

potential temperature perturbation

θ̂ amplitude of potential temperature perturbation

θ0 constant reference potential temperature,

potential temperature at the height of the

roughness length

θ c potential temperature in well-mixed region

between split streamlines in flow over a barrier

θ e equivalent potential temperature

θ∗e equivalent potential temperature if air is saturatedat its current temperature and pressure

θ∗e mean (environmental) equivalent potentialtemperature if air is saturated at its current

temperature and pressure

θ ep pseudoequivalent potential temperature

θ v virtual potential temperature

θv mean (environmental) virtual potential

temperature

θ ′v virtual potential temperature perturbation

θw wet-bulb potential temperature

θρ density potential temperature

θρ mean (environmental) density potential

temperature

θ′ρ density potential temperature perturbation

κ wave vector

κ thermal diffusivity

κe moisture diffusivity

λ longitude, wavelength

λx zonal wavelength

λz vertical wavelength

µ a real number

ν kinematic viscosity

π 3.141 592 65, Exner function

xviii LIST OF SYMBOLS

π mean (environmental) Exner function

π′

perturbation Exner function

ρ air density

ρ0 constant reference density

ρa density of an adiabatic reference state

ρd density of dry air

ρi density of ice hydrometeor

ρv density of water vapor

ρ mean (environmental) air density

ρ′

air density perturbation

σ static stability parameter, growth rate of

isentropic surface

τ lifetime of a convective cell

φ latitude, radar beam elevation angle, phase of

radar transmission

� geopotential

� mean geopotential

�′

geopotential perturbation

�′i imaginary part of the geopotential perturbation

�′r real part of geopotential perturbation

�′ * complex conjugate of the geopotential

perturbation

ψ streamfunction

ψ0 angular constant designating the orientation of

the ageostrophic wind at the start of the inertial

oscillation that leads to the nocturnal low-level

wind maximum

ψ mean streamfunction

ψ′

streamfunction perturbation

ψ̂ complex amplitude of streamfunction

perturbation

� Earth’s angular velocity vector

� angular rotation rate of Earth, intrinsic frequency

ω relative vorticity vector

ωh horizontal vorticity vector

ω frequency

ωc crosswise vorticity component

ωk frequency of kth mode

ωs streamwise vorticity component

A area of an arbitrary surface bounded by the circuit

about which circulation is computed

Ae projection of A onto the equatorial plane

a radius of Earth, shape parameter for terrain

profile

B buoyancy

Bu Burger number

C circulation, condensation rate, speed of bore

relative to upstream density current, radar

constant

Ca absolute circulation

Cp heat capacity at constant pressure

c storm motion vector

cg group velocity

c phase speed, speed of light

c∗ complex conjugate of the phase speed

cd drag coefficient

ce bulk transfer coefficient for moisture

cgx zonal group velocity component

cgz vertical group velocity component

ch bulk transfer coefficient for heat

ci imaginary part of phase speed

cl specific heat of liquid water for a constant

pressure process

cp specific heat for a constant-pressure process

cpd specific heat at constant-pressure for dry air

cpv specific heat at constant-pressure for water vapor

cr real part of phase speed

cv specific heat for a constant-volume process

cvd specific heat at constant-volume for dry air

cvv specific heat at constant volume for water vapor

D characteristic depth scale, resultant deformation,

depth of wave duct, depth of fluid layer, depth

of outflow, duration of precipitation, hailstone

diameter

D1 stretching deformation

D2 shearing deformation

d depth of control volume

dA element of an arbitrary surface having an area A

dl element of a circuit about which circulation isevaluated

E evaporation rate, precipitation efficiency

e vapor pressure, Euler’s number

LIST OF SYMBOLS xix

e mean vapor pressure, turbulent kinetic energy

eij deformation tensor

es saturation vapor pressure

F viscous force

Fh sum of horizontal forces acting on an air parcel

Fu viscous force acting on u

Fv viscous force acting on v, sum of vertical forces

acting on an air parcel

Fw viscous force acting on w

Fr Froude number

Frm mountain Froude number

f Coriolis parameter, frequency

f0 constant reference Coriolis parameter

g gravitational acceleration vector

g gravitational acceleration

g′

reduced gravity

H scale height of atmosphere, undisturbed depth of

fluid layer, far-field depth of cold pool

H0 original height of dividing streamline

H1 nadir height of dividing streamline

h specific enthalphy, height above ground

h0 depth of stable layer

h1 depth of bore

hI inertial height scale

hm height of mountain summit

ht height of terrain

Iδ vertical integral of the displacement of potential

temperature surfaces

i unit vector in positive x direction

i√−1

j unit vector in positive y direction

Ke eddy diffusivity for moisture

Kh eddy diffusivity for heat

Km eddy viscosity

k unit vector in positive z direction

k zonal wavenumber, von Karman’s constant, wave

mode

KE kinetic energy

LR Rossby radius of deformation

LRm mountain Rossby radius of deformation

Lx distance between mountain crests

l meridional wavenumber, mixing length,

cross-gap length scale, Scorer parameter

lf specific latent heat of fusion

ls specific latent heat of sublimation

lv specific latent heat of vaporization

M angular momentum, absolute (or pseudoangular)

momentum

M mean angular momentum

M′

angular momentum perturbation

Mg geostrophic absolute (or geostrophic

pseudoangular) momentum

m vertical wavenumber

N Brunt-Väisälä frequency, refractivity

Nm moist Brunt-Väisälä frequency

N0 constant Brunt-Väisälä frequency

n unit vector that points to the left of the horizontalwind velocity

n coordinate in the n direction, an integer,refractive index

Pr received backscattered power

p pressure

p mean (environmental) pressure

p0 reference pressure

pd pressure of dry air

p∗ saturation pressure

p′

pressure perturbation

p̂ amplitude of pressure perturbation

p′b buoyancy pressure perturbation

p′d dynamic pressure perturbation

p′h hydrostatic pressure perturbation

p′nh nonhydrostatic pressure perturbation

p′dl linear dynamic pressure perturbation

p′dnl nonlinear dynamic pressure perturbation

p̂ complex amplitude of pressure perturbation

p∞ ambient far-field pressure away from a tornado

PV Ertel’s potential vorticity

PVg geostrophic potential vorticity

Q heating rate

Qe surface latent heat flux

xx LIST OF SYMBOLS

Qh surface sensible heat flux

Qg ground heat flux

q specific heating rate

R gas constant, radius of circulation circuit,

reflection coefficient, rainfall rate

R∗ complex conjugate of reflection coefficient

Rd gas constant for dry air

Rf flux Richardson number

Rn net radiation

Rt radius of curvature of a trajectory

Rv gas constant for water vapor

r position vector

r distance to center of Earth, radial coordinate,

range to radar target, linear correlation

coefficient, aspect ratio of a mountain

rh hydrometeor mixing ratio

rt total water mixing ratio

rv water vapor mixing ratio

rv0 water vapor mixing ratio at the height of the

roughness length

rv mean water vapor mixing ratio

rva mean water vapor mixing ratio at anemometer

level

r′v water vapor mixing ratio perturbation

rvs saturation water vapor mixing ratio

Ra Rayleigh number

Rac critical Rayleigh number

Re Reynolds number

Ri Richardson number

Ro Rossby number

RH relative humidity

S mean vertical wind shear vector

S swirl ratio

Si sources and sinks of water vapor

s unit vector that points in the direction of thehorizontal wind velocity

s coordinate in the s direction

T absolute temperature, characteristic timescale

T mean (environmental) absolute temperature

T′

absolute temperature perturbation

T0 constant reference absolute temperature

Td dew-point temperature

Te equivalent temperature

Tv virtual temperature

Tv mean (environmental) virtual temperature

T′v virtual temperature perturbation

Tw wet-bulb temperature

T∗ saturation temperature

Tρ density temperature

t time

U along-gap wind speed

u zonal wind component, radial wind component,

cross-mountain wind component, cross-gap

wind speed

u mean (environmental) zonal wind component

ua mean zonal wind at anemometer level

u′ zonal wind perturbation

û amplitude of zonal wind perturbation

u∗ friction velocity

ua zonal ageostrophic wind component

u0 constant reference zonal wind component, wind

speed far upstream of a mountain

ua0 zonal ageostrophic wind at the start of the inertial

oscillation that leads to the nocturnal low-level

wind maximum

ug zonal geostrophic wind component

ugc along-front geostrophic wind component on cold

side of front

ugw along-front geostrophic wind component on

warm side of front

V characteristic velocity scale, horizontal wind

speed, volume of air

Vg geostrophic wind speed

v wind velocity vector

v mean (environmental) wind velocity vector

v′ perturbation wind velocity vector

va ageostrophic wind vector

va0 ageostrophic wind vector at the start of theinertial oscillation that leads to the nocturnal

low-level wind maximum

vg geostrophic wind vector

LIST OF SYMBOLS xxi

vh horizontal wind velocity vector

vT thermal wind vector

v meridional wind component, tangential wind

component, mountain-parallel wind

component

v mean (environmental) meridional wind

component

va mean meridional wind at anemometer level

v′ meridional wind perturbation

vR radial velocity

va meridional ageostrophic wind component

va0 meridional ageostrophic wind at the start of the

inertial oscillation that leads to the nocturnal

low-level wind maximum

vg meridional geostrophic wind component

vt hydrometeor fall speed

W work, width, sum of vertical velocity of air plus

hydrometeor fall speed

W↓ work required to displace parcel downward

W↑ work required to displace parcel upward

w vertical wind component

w mean vertical wind component

w′ vertical velocity perturbation

ŵ amplitude of vertical velocity perturbation

w̃ complex amplitude of the vertical velocity

perturbation

w̃k kth mode of the complex amplitude of the vertical

velocity perturbation

w̃ki imaginary part of the kth mode of the complex

amplitude of the vertical velocity perturbation

w̃kr real part of the kth mode of the complex

amplitude of the vertical velocity perturbation

x coordinate in the i direction

y coordinate in the j direction

Z impedance, logarithmic reflectivity factor

Zhh reflectivity factor associated with horizontally

polarized transmitted and backscattered pulses

Zvv reflectivity factor associated with vertically

polarized transmitted and backscattered pulses

ZDR differential reflectivity factor

z coordinate in the k direction, reflectivity factor

z′ characteristic distance a parcel travels beforemixing with its surroundings

z0 roughness length, height of a streamline far

upstream of a mountain

zi height of the inversion at the top of the boundary

layer

zinv height of inversion

zr height of interface separating two layers of fluid

PART IGeneral Principles

1What is the Mesoscale?

1.1 Space and time scalesAtmospheric motions occur over a broad continuum ofspace and time scales. The mean free path of molecules(approximately 0.1 µm) and circumference of the earth(approximately 40 000 km) place lower and upper boundson the space scales of motions. The timescales of atmo-spheric motions range from under a second, in the caseof small-scale turbulent motions, to as long as weeks inthe case of planetary-scale Rossby waves. Meteorologicalphenomena having short temporal scales tend to have smallspatial scales, and vice versa; the ratio of horizontal spaceto time scales is of roughly the same order of magnitude formost phenomena (∼10 m s−1) (Figure 1.1).

Before defining the mesoscale it may be easiest first todefine the synoptic scale. Outside of the field of mete-orology, the adjective synoptic (derived from the Greeksynoptikos) refers to a ‘‘summary or general view of awhole.’’ The adjective has a more restrictive meaning tometeorologists, however, in that it refers to large spacescales. The first routinely available weather maps, producedin the late 19th century, were derived from observationsmade in European cities having a relatively coarse character-istic spacing. These early meteorological analyses, referredto as synoptic maps, paved the way for the Norwegiancyclone model, which was developed during and shortlyafter World War I. Because only extratropical cyclones andfronts could be resolved on the early synoptic maps, syn-optic ultimately became a term that referred to large-scaleatmospheric disturbances.

The debut of weather radars in the 1940s enabledphenomena to be observed that were much smaller inscale than the scales of motion represented on synopticweather maps. The term mesoscale appears to have been

Mesoscale Meteorology in Midlatitudes Paul Markowski and Yvette Richardson 2010 John Wiley & Sons, Ltd

introduced by Ligda (1951) in an article reviewing the useof weather radar, in order to describe phenomena smallerthan the synoptic scale but larger than the microscale, a termthat was widely used at the time (and still is) in referenceto phenomena having a scale of a few kilometers or less.1

The upper limit of the mesoscale can therefore be regardedas being roughly the limit of resolvability of a disturbanceby an observing network approximately as dense as thatpresent when the first synoptic charts became available,that is, of the order of 1000 km.

At least a dozen different length scale limits for themesoscale have been broached since Ligda’s article. Themost popular bounds are those proposed by Orlanski(1975) and Fujita (1981).2 Orlanski defined the mesoscaleas ranging from 2 to 2000 km, with subclassifications ofmeso-α, meso-β, and meso-γ scales referring to horizontalscales of 200–2000 km, 20–200 km, and 2–20 km, respec-tively (Figure 1.1). Orlanski defined phenomena havingscales smaller than 2 km as microscale phenomena, andthose having scales larger than 2000 km as macroscale phe-nomena. Fujita (1981) proposed a much narrower rangeof length scales in his definition of mesoscale, where themesoscale ranged from 4 to 400 km, with subclassifica-tions of meso-α and meso-β scales referring to horizontalscales of 40–400 km and 4–40 km, respectively (Figure 1.1).

1 According to Ligda (1951), the first radar-detected precipitation areawas a thunderstorm observed using a 10-cm radar in England on20 February 1941. Organized atmospheric science research using radarswas delayed until after World War II, however, given the importanceof the relatively new technology to military interests and the secrecysurrounding radar development.2 In addition to Orlanski and Fujita, scale classifications and/or subclas-sifications also have been introduced by Petterssen (1956), Byers (1959),Tepper (1959), Ogura (1963), and Agee et al. (1976), among others.

4 WHAT IS THE MESOSCALE?

tim

esc

ale

horizontal length scale

20 m 200 m 2 km 20 km 200 km 2000 km

turbulence

dust devilsthermals

tornadoes

10000 km

1 second

1 hour

1 minute

1 day

1 month

micro αscale

micro βscale

micro γscale

meso α scale

meso β scale

meso γscale

macro β scale

macro α scale

large tornadoes

thunderstorms

urban effects

mountain & lakedisturbances

fronts (along-front dimension)

hurricanes

baroclinic waves

“long waves”

cir

cu

mfe

ren

ce

of

Ea

rth

40

00

0 k

m

s lope

~10

m s

-1

Orlanski (1975)

Fujita (1981)

~2π / N

~2π / f

miso α scale

miso β scale

moso αscale

maso β scale

meso α scale

meso β scale

maso α scale

shortgravity waves

convectivesystems

Figure 1.1 Scale definitions and the characteristic time and horizontal length scales of a variety of atmosphericphenomena. Orlanski’s (1975) and Fujita’s (1981) classification schemes are also indicated.

Fujita’s overall scheme proposed classifications spanningtwo orders of magnitude each; in addition to the mesoscale,Fujita proposed a 4 mm–40 cm musoscale, a 40 cm–40 mmososcale, a 40 m–4 km misoscale, and a 400–40 000 kmmasoscale (the vowels A, E, I, O, and U appear in alpha-betical order in each scale name, ranging from large scalesto small scales). As was the case for Fujita’s mesoscale,each of the other scales in his classification scheme wassubdivided into α and β scales spanning one order ofmagnitude.

The specification of the upper and lower limits ofthe mesoscale does have some dynamical basis, althoughperhaps only coincidentally. The mesoscale can be viewedas an intermediate range of scales on which few, if any,simplifications to the governing equations can be made, atleast not simplifications that can be applied to all mesoscalephenomena.3 For example, on the synoptic scale, several

3 This is essentially the same point as made by Doswell (1987).

terms in the governing equations can safely be disregardedowing to their relative unimportance on that scale, suchas vertical accelerations and advection by the ageostrophicwind. Likewise, on the microscale, different terms in thegoverning equations can often be neglected, such as theCoriolis force and even the horizontal pressure gradientforce on occasion. On the mesoscale, however, the full com-plexity of the unsimplified governing equations comes intoplay. For example, a long-lived mesoscale convective systemtypically contains large pressure gradients and horizontaland vertical accelerations of air, and regions of substantiallatent heating and cooling and associated positive andnegative buoyancy, with the latent heating and coolingprofiles being sensitive to microphysical processes. Yet eventhe Coriolis force and radiative transfer effects have beenshown to influence the structure and evolution of thesesystems.

The mesoscale also can be viewed as the scale on whichmotions are driven by a variety of mechanisms ratherthan by a single dominant instability, as is the case on

DYNAMICAL DISTINCTIONS BETWEEN THE MESOSCALE AND SYNOPTIC SCALE 5

the synoptic scale in midlatitudes.4 Mesoscale phenomenacan be either entirely topographically forced or drivenby any one of or a combination of the wide variety ofinstabilities that operate on the mesoscale, such as thermalinstability, symmetric instability, barotropic instability, andKelvin-Helmholtz instability, to name a few. The dominantinstability on a given day depends on the local state ofthe atmosphere on that day (which may be heavily influ-enced by synoptic-scale motions). In contrast, midlatitudesynoptic-scale motions are arguably solely driven by baro-clinic instability; extratropical cyclones are the dominantweather system of midlatitudes on the synoptic scale. Baro-clinic instability is most likely to be realized by disturbanceshaving a horizontal wavelength roughly three times theRossby radius of deformation, LR, given by LR = NH/f ,where N, H, and f are the Brunt-Väisälä frequency, scaleheight of the atmosphere, and Coriolis parameter, respec-tively.5 Typically, LR is in the range of 1000–1500 km. Ineffect, the scale of the extratropical cyclone can be seen asdefining what synoptic scale means in midlatitudes.

In contrast to the timescales on which extratropicalcyclones develop, mesoscale phenomena tend to be shorterlived and also are associated with shorter Lagrangiantimescales (the amount of time required for an air parcel topass through the phenomenon). The Lagrangian timescalesof mesoscale phenomena range from the period of a purebuoyancy oscillation, equal to 2π/N or roughly 10 minuteson average, to a pendulum day, equal to 2π/f or roughly17 hours in midlatitudes. The former timescale could beassociated with simple gravity wave motions, whereas thelatter timescale characterizes inertial oscillations, such asthe oscillation of the low-level ageostrophic wind com-ponent that gives rise to the low-level wind maximumfrequently observed near the top of nocturnal boundarylayers.

The aforementioned continuum of scales of atmosphericmotions and associated pressure, temperature, and mois-ture variations is evident in analyses of meteorological

4 See, for example, Emanuel (1986).5 In addition to being related to the wavelength that maximizes thegrowth rate of baroclinic instability, LR also is important in the problemof geostrophic adjustment. Geostrophic adjustment is achieved by rela-tively fast-moving gravity waves. The horizontal scale of the influence ofthe gravity waves is dictated by LR, which physically can be thought ofas the distance a gravity wave can propagate under the influence of theCoriolis force before the velocity vector is rotated so that it is normal tothe pressure gradient, at which point the Coriolis and pressure gradientforces balance each other. For phenomena having a horizontal scaleapproximately equal to LR, both the velocity and pressure fields adjustin significant ways to maintain/establish a state of balance between themomentum and mass fields. On scales much less than (greater than) LR,the pressure (velocity) field adjusts to the velocity (pressure) field duringthe geostrophic adjustment process.

variables. Figure 1.2 presents one of Fujita’s manual anal-yses (i.e., a hand-drawn, subjective analysis) of sea levelpressure and temperature during an episode of severethunderstorms.6 Pressure and temperature anomalies areevident on a range of scales: for example, a synoptic-scalelow-pressure center is analyzed, as are smaller-scale highsand lows associated with the convective storms. The magni-tude of the horizontal pressure and temperature gradients,implied by the spacing of the isobars and isotherms, respec-tively, varies by an order of magnitude or more within thedomain shown.

The various scales of motion or scales of atmosphericvariability can be made more readily apparent by way offilters that preferentially damp select wavelengths whileretaining others. For example, a low-pass filter can be usedto remove relatively small scales from an analysis (low-passrefers to the fact that low-frequency [large-wavelength]features are retained in the analysis). A band-pass filter canbe used to suppress scales that fall outside of an intermediaterange. Thus, a low-pass filter can be used to expose synoptic-scale motions or variability and a band-pass filter can beused to expose mesoscale motions. (A high-pass filter wouldbe used to suppress all but the shortest wavelengths presentin a dataset; such filters are rarely used because the smallestscales are the ones that are most poorly resolved andcontain a large noise component.) The results of suchfiltering operations are shown in Figure 1.3, which servesas an example of how a meteorological field can be viewedas having components spanning a range of scales. The totaltemperature field comprises a synoptic-scale temperaturefield having a southward-directed temperature gradientplus mesoscale temperature perturbations associated withthunderstorm outflow.

1.2 Dynamical distinctionsbetween the mesoscaleand synoptic scale

1.2.1 Gradient wind balance

On the synoptic scale, phenomena tend to be characterizedby a near balance of the Coriolis and pressure gradient forces(i.e., geostrophic balance) for straight flow, so accelerationsof air parcels and ageostrophic motions tend to be verysmall. For curved flow, the imbalance between these forceson the synoptic scale results in a centripetal accelerationsuch that the flow remains nearly parallel to the curved

6 Fujita called these mesoscale meteorological analyses mesoanalyses. Theanalyses he published over the span of roughly five decades are widelyregarded as masterpieces.

6 WHAT IS THE MESOSCALE?

Figure 1.2 Sea-level pressure (black contours) and temperature (red contours) analysis at 0200 CST 25 June 1953. A squallline was in progress in northern Kansas, eastern Nebraska, and Iowa. (From Fujita [1992].)

isobars (i.e., gradient wind balance). Gradient wind balanceis often a poor approximation to the air flow on themesoscale. On the mesoscale, pressure gradients can beconsiderably larger than on the synoptic scale, whereasthe Coriolis acceleration (proportional to wind velocity) isof similar magnitude to that of the synoptic scale. Thus,mesoscale systems are often characterized by large windaccelerations and large ageostrophic motions.

As scales decrease below ∼1000 km the Coriolis accel-eration becomes decreasingly important compared withthe pressure gradient force, and as scales increase beyond∼1000 km ageostrophic motions become decreasingly sig-nificant. Let us consider a scale analysis of the horizontalmomentum equation (the x equation, without loss ofgenerality):

du

dt= − 1

ρ

∂p

∂x+ f v + Fu, (1.1)

where u, v, ρ, p, f , d/dt, and Fu are the zonal windspeed, meridional wind speed, air density, pressure, Coriolis

parameter, Lagrangian time derivative, and viscous effectsacting on u, respectively. We shall neglect Fu for now, butwe shall find later that effects associated with the Fu termare often important.

On the synoptic scale and mesoscale, for O(v) ∼10 m s−1, the Coriolis acceleration f v is of order

O(f v) ∼ (10−4 s−1) (10 m s−1) ∼ 10−3 m s−2.

On the synoptic scale, the pressure gradient force has a scaleof

O

(− 1

ρ

∂p

∂x

)∼ 1

1 kg m−310 mb

1000 km∼ 10−3 m s−2;

thus, the Coriolis and pressure gradient forces are of simi-lar scales and, in the absence of significant flow curvature,we can infer that accelerations (du/dt) are small. Fur-thermore, because v = vg + va and vg = 1ρf ∂p∂x , where vgand va are the geostrophic and ageostrophic meridionalwinds, respectively, (1.1) can be written as (ignoring Fu)