Introduction to Mesoscale Meteorology

Overview

• Scale Definitions– Synoptic

• Synoptic derived from Greek “synoptikos” meaning general view of the whole. Also has grown to imply “at the same time” or “simultaneous”.

– Synoptic Scale • The scales of fronts and cyclones studied by the early

Norwegian scientists. The classic synoptic scale are the time and space scales resolved by observations taken at major European cities having a mean spacing of about 100 km. Hence weather systems having scales of a few hundred kilometers or more and time scales of a few days are generally what is accepted to be “synoptic scale” phenomena.

Overview

• Scale Definitions– Cumulus

• Defined by the rise of RADAR meteorology in the late 1940’s to be the scale of individual thunderstorm and cumulus cell echoes, this became the second important scale of meteorology research. This scale is on the order of a couple of kilometers to about 50 km and time scales of a few minutes to several hours.

Overview

• Scale Definitions– Mesoscale (original definition)

• Coined by Lidga (1951), mesoscales are the “Middle Scales” between synoptic scale and cumulus scale. This original definition hence referred to weather phenomena of scales between what were thought to be the two primary energy containing scales of cumulus and synoptic scale. The Modern Definition is much more robust.

Overview

• Scale Definitions– Mesoscale (modern definition)

• Orlanski (1975) proposed a new set of scales (ignoring synoptic and cumulus) that include the micro-, meso- and macro- scales. Figure 1 depicts these three definitions. All three definitions have gained wide acceptance, despite an even newer proposal by Fugita (1981). His definition of the “mesoscale” was scales between 2 km and 2000 km. Scales larger than 2000 km are “macroscale” and scales smaller than 2 km are “microscale”

Mesoscale Over The

Years

Overview

• Scale Definitions– Mesoscale (modern)

• Orlanski divides the “mesoscale” into three “sub-mesoscales”:1. Meso- : 2-20 km2. Meso- : 20-200 km3. Meso- : 200-200 km

• We will attach a physical significance to these three mesoscales.

Equations of Motion

ui

t u j

ui

x j

i, j ,k f juk 1pxi

i3g Momentum/Density Form

ui

t u j

ui

x j

i, j ,k f juk xi

i3g Momentum/Entropy Form

ui

t u j

ui

x j

i, j ,k f juk '

xi

i3g '

o

Quasi-Boussinesq Momentum/Entropy Form

ui

t i, j ,k ( j f j )uk kxi

xi

i3g Bernoulli Form

ui

t li kxi

pxi

n Lamb Vector Form

212 ik u , ,

ji i j k

k

ux

, ,i j k j kA B A B

, ,

0 i=j, or j=k, or i=k1 i,j,k are 1,2,3 or 2,3,1 or 3,1,21 i,j,k are 3,2,1 or 2,1,3 or 1,3,2

i j k

ififif

1000p

Rc

ppc

mb

pxi

n xi

i3g

Force Balance in Equation of Motion

ui

t li kxi

pxi

n

| |

Nonhydrostatic Pressure Gradient

| |________________

Inertial Balance

Rotational Irrotational

Lamb Vector Kinetic Energy Gradient

Inertial acceleration Real force acceleration| |

hydrostatic imbalance

Physical Significance of Mesoscale

• Two Major Categories of Dynamic Force Balances Result

1. Hydrostatic: Gravity versus Pressure Gradient 2. Inertial: Inertial Force Versus Gravity

a) Geostrophic (Horizontal Pressure Gradient versus vertical Coriolis effect)

b) Cyclostrophic (Pressure Gradient versus rotational and irrotational inertial)

c) Gradient (Horizontal Pressure gradient versus all inertial)

Perturbations from Balance

• For stable balance, i.e. stability restores balance, perturbations initiate oscillations that result in waves

• For unstable balance, perturbations produce a growing disturbance

Perturbations from Hydrostatic Balance

• Perturbations from stable balance lead to:– Gravity or Buoyancy waves– Horizontal phase speed is

• Perturbations from unstable balance hydrostatic balance (static instability) lead to:– Convection

2z

go

N L gck z

Perturbations from Geostrophic Balance

• Stability produces Balance:– Oscillation frequency is f– Wave speed is on order of :

• Unstable balance produces:– Inertial Instability

2x

ifLfc

k

If both hydrostatic and inertial balances occur and the flow is perturbed, what is the result?

• Depends on which adjustment dominates.• Determine dominant adjustment from ratio

of gravity wave phase speed to inertial wave phase speed:

g zo

i h

c L NRc L f

The Rossby Radius of Deformation

• Scale at which there is equal inertial and gravity wave response

• The definition of Rossby Radius is:

g ZR

c L N HNLf f f

Rossby Radius for axi-symmetric vortex having tangential wind

“V” and Radius “R”

1

1 22

2

gR

cL

Vf fR

Scale Based on Physical Mechanism

• Small Scales– Frequency of gravity waves, ie Brunt-Visalia

Frequency, larger than frequency of inertial waves– Tendency toward hydrostatic balance with “g”

dominate

• Large Scales– Frequency of inertial wave involving Coriolis larger

than gravity wave– Balance against inertial acceleration dominates

Back to Mesoscale Definitions

• At middle latitudes (40 N) :

• For a disturbance depth of 7 km:

• Hence Rossby radius is typically:

4 110f s

120gc ms

200RL km

• Scales from 2-20 km• Disturbances characterized by Gravity (Buoyancy)

Waves (stable) or Deep Convection (unstable).• Coriolis effect generally negligible, although local

inertial effects can arise to change character of disturbances (i.e. rotating thunderstorms, tornadoes, dust devils, etc)

Meso

Roll convection

Open and Closed

Cells with Lee Eddies

Open and Closed Cells

Lake Effect Clouds

Ship Tracks In stratocumulus off Oregon

Sea Fog

Valley Fog

Trapped Lee Waves in Ireland

Mountain Wave Disturbances

Orographic Precipitation

Supercells

Tornado

• Scales of 20-200 km• Less than but near to Rossby Radius• Gravity (Buoyancy) Waves govern system

evolution and propagate relative to the wind• Inertial oscillation important to wave

dynamics, i.e. Gravity-Inertia Waves

Meso

Shore Parallel Lake Effect Band

Actinae

Actinae.fli

Sea Breeze Convection

Seabreeze

Meso-beta squall lines

• Scales of 200-2000 km • Scales greater than but near to Rossby Radius of

deformation• Characterized by Geostrophic Balance• Geostrophic disturbance determines evolution of

the system• Vertical ageostrophic motions driven by

geostrophic disturbance, i.e. quasi-geostrophic dynamics

Meso-

Squall Line

MCC

MCC

ITCZ Cluster

Tropical Cyclone

Exceptions to these rules• These above rules are an approximate description

of how the three mesoscales divide typical disturbances.

• Changes in latitude mean changes in attitude! As we move to lower latitudes Coriolis effect decreases and the Rossby Radius scale increases, going to infinity at the equator! Hence relative to the Earth’s inertial effect, all disturbances around the equator are dynamically small and are governed by gravity (buoyancy) waves. Hence, the Kelvin wave, a gravity type wave, is a global scale equatorial disturbance.

• Deeper disturbances lead to larger gravity wave phase speeds and so have larger Rossby Radius and vise versa for shallow disturbances. Hence, the depth of the disturbance will affect its governing dynamics, the deeper disturbance less likely to be inertially balanced and the shallower disturbances more likely. Some examples:– Sea Breeze without deep convection: shallow, likely to

achieve significant geostrophic balance– Sea Breeze with Deep convection: Deep, not likely to

achieve inertial balance– Mesoscale Convective Complex: Deep convection parts

of the system clearly less than Rossby Radius. Stratiform anvil has large horizontal scale shallow melting layer that may excite inertially balanced disturbance, ie an MCV (mesoscale convective vortex).

• Rotation induced in the system may locally shrink the Rossby Radius, making the system dynamically large even on meso-beta scales. For example:– Mesocyclone or rotating thunderstorm with its forward

flanking gust front and rear flanking gust fronts actually become quasi balanced and evolve very similar to a developing quasi-geostrophic baroclinic cyclone with warm and cold fronts respectively. This makes the supercell thunderstorm long lived.

– The tropical cyclone eye wall becomes inertially stable from the strong storm rotation giving rise to Rossby wave disturbances (relative to cyclone rotation instead of Coriolis) that move around the eye, and play significant roles in horizontal momentum transport through their tilt!

– Tornadoes survive relatively long periods for their size because of their inertial balance and locally very small Rossby Radius of Deformation.

Genesis of Fefa

Special Observation and Analysis Problems of the Mesoscale

• Synoptic observation systems have horizontal resolutions of 100 km and 1 hour at the surface and 400 km and 12 hours aloft and are clearly inadequate to capture all but the upper end of the meso- .

• The dynamics of mesoscale disturbances contain important non-balanced or transient features that propagate rapidly.

• The systems are highly three-dimensional so that the vertical structure is equally important to the horizontal structure.

• Mesoscale disturbances are more likely to be a hybrid of several dynamic entities interacting together to maintain the system.

• Process Interaction is especially significant. Such processes include microphysical and radiative transfer interactions.

• Scale Interactions and particularly interactions across the Rossby Radius of Deformation are basic to the mesoscale problem

How Can we Deal with This?• Use observations as clues to the analysis, and do

not expect the data to ever be sufficient to reveal the process behind the observations.

• Attach a strong dynamical model to the observations to fill in the gaps. To the extent that the model reproduces the observations at points where it is coincident with the observations, it gains credibility.

• Study the model to understand the dynamics. If the model is consistent with the few observations, then the model can be used (always with caution) to reveal the dynamics of the system.

What is a Model?

• Models range from simple to complex.• Simple model: Quasi-Geostrophic model:

positive vorticity advection results in upward vertical motion. Its so simple we can do it in our heads! But it has many approximations and is likely to miss features that can be represented in more complex models.

• Primitive Equation Forecast Model: Such as the Eta model, or AVN model or ECMWF model. Must be solved on the computer but much more precise than the simple PVA model. Why would anyone look at a 500mb map predicted by the Eta model and then disagree with its vertical motion pattern because it doesn’t obey the mentally tractable PVA model? Some people do!

• Research Models: More precise physics, too big to execute in real time but able to provide a deeper understanding of specific processes causing an observed event.

• These models, used in case study or idealized mode provide our basic understanding that we use to construct truncated models that can run in real time.