THEME : Transport Processes in the Atmosphere Environmental Fluid Dynamics II: Atmospheric Portion

THEME : Transport Processes in the Atmosphere Environmental Fluid Dynamics II: Atmospheric Portion

Aim : To provide a rudimentary overview of the nature and role of the major atmospheric transport processes.

The “atmospheric” component of EFD II is designed to meet the following three criteria :

- builds naturally upon the material in EFD I, - theme of general “env. sci.” interest, and

- complements “atmospheric dynamics” lectures.

THEME : Transport Processes in the Atmosphere Environmental Fluid Dynamics II: Atmospheric Portion

OUTLINE

I INTRODUCTION

Motivation (- significance of theme, sample issues)

II PRELIMINARY THEORETICAL CONSIDERATIONS

Basic & Extended Equation(s) Illustrative flow examples

Scale Analysis & Scale Contraction Integral relation

III SELECTED PHENOMENA / PROCESSES

Troposphere: Conveyor Belts, ....

S-T Interface Exchange & Mixing

Stratosphere : Brewer Dobson Circulation Sudden Stratospheric Warming Ozone hole

MOTIVATION: Why this theme ?

Large-scale atmospheric flow is irregular, complex, and chaotic with a resulting

rich range of evolving and intricate patterns and structures for tracers and constituents.

MOTIVATION: Why this theme ?

Influences a variety of processes /cycles.

Seasonal, inter-annual and longer time-scales the magnitude and

distribution of the atmospheric constituents exerts a major control upon - the atmosphere’s radiative heating and transfer properties, and thereby

has a direct influence upon climate,

- air quality characteristics both near the earth’s surface and in the free atmosphere, and thereby has a direct influence upon the habitability and quality of life,

- the impact of volcanic eruptions upon weather and climate, and likewise the levels of ozone in the stratosphere,

- the global cycling of greenhouse gases (including water vapour, carbon dioxide, methane and ozone), is linked to their atmospheric residence time and

geographical distribution,

Central to the atmospheric component of the various bio-geochemical cycles

Assimilation of CO

₂

Troposphere Stratosphere

ECMWF assimilation / analyses

On day-to-day time-scales the large-scale flow strongly influences

- the space-time distribution of atmospheric pollutants emitted from localized sources, and there are a raft of international conventions that require a better quantification of trans-boundary fluxes,

- the transport between their source and deposition of constituents recorded in ice cores and sediments

.

Distribution and evolution of atmospheric quasi-tracers (humidity, ozone,..) provides additional information on the atmospheric state and can thereby help to determine the initial state of the atmosphere for numerical weather prediction

A key atmospheric variable (potential vorticity) that is central to the

Note that

- distribution of constituents in the stratosphere is somewhat counter-intuitive, - maintenance of the ozone hole is indicative of a lack of meridional transport, - evidence suggests that the tropopause is a “leaky” barrier to transport despite its alleged impermeability,

-the time-mean tropospheric circulation is not an effective transporter, but

extratropical cyclones accomplish significant poleward plus vertical transport

along gently sloping isentropic surfaces.

II. PRELIMINARY THEORETICAL CONSIDERATIONS

2.1 GENERALIZED SPECIES / TRACER EQUATION 2.1.1 The BASIC EQUATION

A generalised equation governing the development of the mixing ratio / concentration of an atmospheric chemical species / constituent takes the form,

DC/Dt = κ∇²C + R + S + G (1)

where κ is the kinematic diffusivity of the constituent, R is the chemical reaction term,

S is the source (or sink) term, and

G is the transfer rate of the species between the gas and hydrometer phase.

DC/Dt is the Lagrangian rate of change of the constituent C, and DC/Dt = {∂C/∂t + V. Ñ C},

where V is the velocity field (possibly incorporating a fall-speed V*).

2.1.2 THE EXTENDED EQUATION

Equation (1) applies to individual air parcels. However it is often convenient to refer to mean and deviation components. (The mean value, [C], can refer to the space and/ or time average value or to the resolved and non-resolved parts).

deviation components,

C = [C] + C’ , V = [V] + V’,

S = [S] + S’, …. etc.

On substitution into Eq. (1) it follows that

D[C]/Dt = κ∇²[C] + [R] + [S] + [G] + ∇. [V’C’] (2)

The additional term, ∇.[V’C’], denotes the contribution to the mean constituent’s evolution by the deviation (or unresolved) components. It is in effect a flux divergence engendered by the turbulent (or sub-scale) flow. For smaller scale features in the atmospheric boundary layer it is often represented by a Fickian diffusion term, but in general it requires

a more sophisticated turbulent closure formulation.

Other representations of Equation (2) take into account the quasi-horizontal nature of large scale atmospheric flow, and

- split the advection and the flux divergence into horizontal and vertical components,

- examine the change of C on isentropic surfaces (- useful if the process is quasi-adiabatic)

2.1.3 SIMPLIFIED SETTINGS

Consider the simpler advection-diffusion equation,

∂C/∂t + V. ∇C = κ∇²C

and assume a constant coefficient of diffusivity, and two dimensional flow.

To explore the possible influence of advection and diffusion consider their effects separately.

(a) Advection

Here we assume there is no diffusion,

i.e. ∂C/∂t + V. ∇C = 0

stipulate the background flow and consider the evolution of the (perfect) tracer under the influence of the flow for two flow settings deemed to be generic.

SETTING I

A “square-wave” tracer distribution immersed in a pure deformation field.

Deformation field (u = - α x; v = + α y)

Tracer Initial dimensions of square wave (wavelength L; amplitude A) .

Under the influence of the deformation field the wave amplitude increases quasi-linearly (- in essence a stretching) whilst

its wavelength decreases uniformly (in essence a compression) to yield a so-called tendril pattern.

SETTING II :

A “line segment” tracer distribution immersed in a “Rankine-type” vortex.

Vortex azimuthal velocity: v = ω r for r < a v = (ωa²)/ r for r > a

Tracer Initial location of line segment

(aligned radially outward from the centre)

Under the influence of the vortex flow the tracer takes on a whorl (spiral) form and the spacing of the striations evolve systematically in time.

(b) Diffusion

Consider the evolution of a tracer in the half-space (y > 0) with no ambient flow, and with the initial tracer distribution such that C = 0 at t = 0 for y >o

and subsequently C = C₀ at y = 0 for t > 0 The tracer satisfies the diffusion equation

∂C/∂t = κ∇²C

The resulting solution takes the form presented in EFD I.

2.1.4 SCALE ANALYSIS

Consider again the advection-diffusion equation,

∂C/∂t + V. ∇C = κ∇²C

O.O.M (CU/L) (κC/L²)

where U and L are characteristic velocity and length scales.

Thus the dimensionless number representing the ratio of the advective and diffusive effects is the Peclet Number, Pe,

Pe = UL/κ ,

In effect if Pe >> 1 then advection dominates, and if Pe << 1 then diffusion dominates

Likewise, unless there is a scale contraction, the appropriate time-scale is T = L² /κ

Note that in the atmosphere the molecular diffusivity of most gaseous chemical species varies from 10^-5 m² s^-1 in the troposphere

to 10^-3 m² s^-1 in the upper stratosphere

(and higher values in the mesosphere)

Thus for significant diffusive mixing on length scales of say 100m would require (c.f. example on diffusion above) ~ 10⁸ s , i.e. a few years.

However if stirring reduces a new length scale (l) for the tracer (c.f. the advection examples above), then the diffusion can become comparable to the advective effect.

One measure of the spatial scaling of a tracer’s structure is defined by ∇ C

and the processes involved in changing that length scale are given by taking the ∇ operator of advection equation

∂C/∂t + V. ∇C = κ∇²C

and for two-dimensional flow the change in the gradient of C along the x-axis is given by, D(∂C/∂x)/Dt = - ( ∂u/∂x ∂C/∂x + ∂v/∂x ∂C/∂y)

Alternatively a convenient scalar measure of the spatial variability of C is given by T = ½{∇ C. ∇ C}

with the time evolution being determined by D(T)/Dt = -∇ C (∇ C.∇)V

= 2T (∂v*/∂n)

Here ω is the vector vorticity, and v* is the velocity component in the n-direction aligned perpendicular to the vector of the gradient of C.

Thus, following a fluid parcel, the three-dimensional vector (∇ C) can be modified in three ways :

- a tendency for "tracer lines" to move against the flow

(c.f. the equation for a material line Da/Dt = {a. ∇ }v ) involving both tilting and compression;

- reorientation by the ambient vorticity; and

- generation / destruction due to spatial variations of diabatic effects.

Likewise the equation for T emphasizes that a decrease of v along the tracer vector connotes enhancement of the gradient.

2.1.5 AREA INTEGRAL

Again consider the simpler advection-diffusion equation,

∂C/∂t + V. ∇ C = κ∇²C

and assume a constant coefficient of diffusivity, and two dimensional flow.

Then the overall spatial variability of C in a given spatial domain S, is given by the integral measure

T

^{= ∫∫}S T ds

It can be shown that

∂

T

^{/∂t = ∫∫}S {( V.∇ C) ∇²C - κ(∇²C)²} ds STIRRING MIXING The first term on the RHS represents stirring and can be positive or negative (but is often locally positive), whilst the second term represents the mixing and is always negative.

In effect stirring tends usually to increase T and mixing to decrease T.

Counter-intuitively the indirect effect of stirring is to decrease T by rendering the mixing more efficient !

III SELECTED PHENOMENA / PROCESSES

(A) Mean Large-scale Flow & Associated Inferences

(B) Some Selected Features

Troposphere: Conveyor Belts, ....

S-T Interface Exchange & Mixing

Stratosphere : Brewer Dobson Circulation

Sudden Stratospheric Warming

Ozone hole

(A) Mean Large-scale Flow & Associated Inferences

Lat. & Time Mean Zonal Wind

Inference: Likelihood of high latitudinal dispersion in vicinity of jet streams

Time-Mean Wind at the upper-troposphere

Time-Mean Wind in the mid-troposphere

Time-Mean Wind in the lower-troposphere

Lat. & Time Mean Zonal Stream Function (sic. meridional circulation)

Inferences: (i) mean meridional circulation (very) weak in (stratosphere) troposphere with with stratosphere / troposphere time scales order of years / weeks

(ii) no mean flux at the sub-tropics / extratropics interface !!

Lat. & Time Mean Isentropic Distribution

Inferences: (i) PV Perspectiove : Tracer & linkage to flow (ii) mean meridional circulation is non-adiabatic

(ii) overworld, middle world, and underworld adiabatic sub-division High PV

High PV

Low PV Low PV

Time Mean “column integrated” ozone distribution

Inferences: (i) tropical source BUT maxima in polar & extratropical regions

(ii) hemispheric asymmetry in seasonal distribution (hint of low polar austral winter values

Further space-time features of ozone distribution

Time Mean “tropospheric” distribution

The Hole

Quasi-instantaneous vertical section Instantaneous regional distribution

(B) Selected Features : Troposphere

Some phenomenological transport features

Inferences: (i) vertical (diabatic) circulations versus quasi-horizontal (sic. isentropic) transport (ii) internal to troposphere versus S-T exchange processes

Semi-quantitative comparison of major transport processes

Instantaneous poleward flux at 850 hPa

q

Significant latitudinal & vertical atmospheric transport accomplished by LSWS & their fine-scale accoutrements

Vertical transport: Parcel paths within a warm conveyor belt.

An example of a coherent ensemble of trajectories (CET).

H. Wernli ECMWF Analysis Data

PV Blocking P<400hPa

-24hrs 700hPa

Q [g/kg] TH [K] PV [pvu]

17K

time

Schwierz et al

Key ensemble of trajectories (- a moist CET) for a particular event

Life-cycle of a coherent air-stream event

Two cases studies of individual events

Case II Case I

Case II

Backward trajectories from a cyclone’s PV- tower

Considerations :

(i) From a PV perspective the evolution of a cyclone to its mature phase requires the vertical superposition of positive PV elements

(ii) Backward trajectories from the tower provides

evidence on the system’s origin, and of the key ingredients

Aerosol band

M. Liniger et al

Geographical origin of band’s air parcels

Development of a fine-scale aerosol band

quasi-isentropioc + decidedly non-isentropic prelude to mixing

(B) Some Selected Features: S-T Interface (Exchange & Mixing)

High PV High PV

Low PV Low PV

High PV High PV

Low PV Low PV

Considerations :

(i) Isentropic S-T exchange conceivable in middle world

with accompanying signatures in the PV, ozone and water vapour patterns on isentropic surfaces

(ii) Actual exchange of ozone / humidity dependent upon

in-situ value, and hence on input from overworld/ upper troposphere.

Overview of Stratosphere-Troposphere

Transport Mechanisms

Signatures on middle-world isentropic surface of

PV Water vapour

The strikingly rich spatial structure of many atmospheric constituents and the resulting enhancement of mixing is, to a significant measure, attributable to LSWS & their fine-scale accoutrements.

Quasi-isentropic prelude to mixing:

Tracer evolution in the lowermost stratosphere over a 24- hour period

Stirring Mechansims

Tendril formation

Whirl /Vortex formation

PV- band Instability

High PV High PV

Low PV Low PV

High PV High PV

Low PV Low PV

Asymmetry of PV Anomalies in the Middle World

Predilection for

- positive anomalies in lowermost stratosphere, - negative anomalies in upper troposphere

Consider fate of PV anomalies of opposite sign

Anomaly linked

velocities Anomaly-induced

velocity

Negative PV anomaly in lowermost stratosphere is extruded into troposphere, and the reverse for a positive PV anomaly.

(B) Some Selected Features:

Stratosphere : Brewer Dobson Circulation Sudden Stratospheric Warming Ozone hole

Schematic of

B-D circulation

The Tape Recorder

Sudden Stratospheric Warming

Time-height section in a polar location during a SSW event

NOTE :

- regular occurrence in the polar Northern Hemisphere

- transient feature (multiple events possible in a single winter

Sequence of charts showing total ozone distribution preceding and during an SSW event

Note :

- indication of upward propagation of wave energy from troposphere, - growth with height of accompanying wind and pressure signal, - stirring can produce a reversal of mean zonal wind !!!

Ozone Hole : A one diagram history

Strong stratospheric Polar Vortex : An isolated chemical crucible

Strong PV gradient serves as a barrier to transport

Problem Sheet

1. A horizontal rectangular fluid strip (200 *10 km) is aligned north-south at 60N with a westerly jet of 50 ms^-1 at its northern end and a 10ms^-1 flow at its southern end. What will be its dimensions (length and lateral scale) four days later ?

2. A square wave with an amplitude of 500km and wavelength 1,000km is immersed in a

deformation field ( u = - αx; v= αy; with α = 10-5 s^-1). What are its dimensions 3 days later ?

3. A line tracer is aligned along a radial direction in a Rankine vortex

( v = ωr for r<a; v = ωa²/r for r>a), and is drawn into a spiral shape by the flow. Show that after a long time (T) the spiral arms are separated by a distance (D) such that

D ~ ( π/ωa²) (r³/T).

4. Perform a scale analysis for Problem 3 of the tracer for large T in the presence of diffusion.

Note that the tracer equation takes the form

∂C/∂t + (v/r)∂C/∂θ = κ{∂²C/∂r² + (1/r²) ∂²C/∂θ ² }.

5. Assume that the mean meridional circulation of the Hadley cell and Brewer Dobson Circulation are respectively about 0.5 and 0.02 ms^-1. Estimate the time-scale of these circulations.