this is the defence presentation for the dissertation dynamics of helical flow structures a...

This is the defence presentation for the dissertation

Dynamics of Helical Flow Structures

A Description of Vortex Formation in Turbulent Fluids

by Nikolaj Nawri

given on 2 April 2003 at the Department of Meteorology of the University of Maryland at College Park.

It is set up for full screen viewing. This can be changed under Slide Show > Set Up Show. In full screen view the speaker notes can be displayed by right-clicking on the

running presentation.

“You don't seriously believe that a theory must restrict itself to observables? Perhaps I did use this sort of

philosophy, but it's nonsense. Only the theory decides what one can observe.”

Albert Einstein to Werner Heisenberg (1926)

Dynamics of Helical Flow Structures

A Description of Vortex Formation in Turbulent Fluids

Nikolaj NawriDepartment of Meteorology

University of MarylandCollege Park, MD, USA

Dynamics of Helical Flow Structures 4

Tornadolike Vortices

A tornadolike vortex is any atmospheric vortex with a diameter of a few hundred metres associated with a

storm system or otherwise driven by larger-scale flow features such as fronts or squall lines.


Tornadolike Vortices

• Tornadolike vortices, by definition, are associated with severe thunderstorms.

• This association of the vortex to the larger-scale circulation of the storm system leads to a strong coupling of a wide range of spatial and temporal scales.

• The complete description of tornadogenesis therefore requires a comprehensive theory of fluid flow including both the intrinsic, universal properties of small scales and a consideration of the energy input on large scales by specific external forcing.


Storm Research

• A common approach employed in observational tornado research is to calculate certain forecast parameters from data of the storm system (obtained prior to tornadogenesis) and to compare their values between tornadic and nontornadic cases.

• Despite unreliable and sparse storm data it appears as if the currently employed parameters are not adequate for reliable predictions of tornadogenesis without high false alarm rates.


Storm Research

• Climatological study of severe thunderstorms by Rasmussen and Blanchard [1998]

• Analyses all of the 0000 UTC soundings from the United States made during the year 1992 that have nonzero CAPE.

• Storms are classified as ordinary thunderstorms, nontornadic supercell thunderstorms, and tornadic supercell thunderstorms.

• Forecast parameters characterising vertical wind shear, static instability, and various combinations thereof are calculated.

Operational Forecast Parameters


Environments of Vortex Formation

• Prior to storm formation atmospheric motion is predominantly horizontal, where over even terrain variability primarily is in the vertical stratification.

• The most favourable large-scale vertical wind profile for the formation of storm rotation is horizontally homogeneous and vertically veering.

• Tornadolike vortices on the other hand are embedded in a storm flow with intense vertical motion and strong horizontal gradients in velocity.

• They commonly form on or near outflow boundaries and shear zones with intense horizontal variability on the vortex scale.


Storm and Vortex Flow

• Since the instabilities of the storm flow are associated with spatial variability on the vortex scale, a separation of the flow into large and small scales is not possible.

• Instead the velocity field is separated into a slowly evolving “background” flow representing the storm motion, and a rapidly evolving perturbation flow representing the tornadolike vortex.


Dynamical Systems Analysis

• From the Fourier transformed equations of motion for storm and vortex flow low-dimensional dynamical systems are derived through spectral truncation.

• In the dynamical system for the rapidly evolving expansion coefficients the slowly evolving expansion coefficients, over short periods of time, are considered to be constant parameters.

• Since the “tornadic” and “nontornadic” parameter regions and the corresponding background flows follow from a bifurcation analysis, an explicit solution of the equations of motion is not necessary to determine the qualitative evolution of the fast flow.


The Quest for New Forecast Parameters

• The dynamically derived parameters are defined as various combinations of the background flow expansion coefficients.

• They are uniquely determined by a given background flow, but given the set of parameters the background flow is not completely specified.

• They are very abstract and unlike the currently employed forecast parameters cannot easily be interpreted in terms of physical concepts such as “static instability” or “vertically veering wind profile.”


Overview

Flow in physical space

• Basic kinematical variables and their importance for the motion of the fluid

• Definition of eddies and flow structures

• Spectral and helical decomposition of storm and vortex flow

• Scales of the storm system


Overview

Flow in phase space

• Three-dimensional dynamical systems

• Equilibria of these systems

• Stability of equilibria and transitions between them

• Interpretation of bifurcations in terms of flow instabilities

Implications for tornado forecasting

Outlook


Flow Properties - Energy

For a given 3D velocity field v = v(t,x) with vorticity = r£ v, the following energy related kinematical variables are defined:

Kinetic energy:

Enstrophy:

Intensity:


Flow Properties - Helicity

To express the relative orientation of velocity and vorticity by scalar fields define:

Alignment:

where

Helicity:


Eddies

• Eddies are defined as positive kinetic energy perturbations from the normal flow state.

• More specifically, helical eddies are defined as positively correlated positive perturbations in intensity and alignment.

• The normal flow state in this study is defined as the relatively slowly evolving velocity field associated with the thunderstorm.


Flow Instability

• The inertial forcing term v £ is responsible for basically all interesting kinematical phenomena in fluid dynamics, in particular for the generation of turbulent velocity perturbations and energy transfer between spectral components of the velocity field.

decreases with increasing alignment.

• The magnitude of inertial forcing


Flow Stability

• While the inertial forcing and shear instabilities are responsible for the onset and maintenance of turbulence, these mechanisms are damaging for an increase in the intensity of large eddies.

• To survive longer as a coherent kinetic energy perturbation, an eddy must acquire kinematical properties that minimise the very forcing term that created the initial disturbance in the normal flow from which it grew.

• Intense velocity perturbations with strong alignment can therefore be expected to be more persistent than velocity perturbations with weak alignment.


Flow Structures

• Since dynamically significant eddies are intense the meaning of helicity for eddies can loosely be expressed as

• Intense and persistent eddies that preserve their qualitative kinematical properties over sufficiently long periods of time to be able to interact with each other are referred to as flow structures.


Helicity Extremisation

For a given intensity helicity is extremised in a finite volume if velocity and vorticity are perfectly aligned:

with a positive constant

These maximally helical flows are called Beltrami flows.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(a)

-pi -pi/2 0 pi/2 pi

-pi

-pi/2

0

pi/2

pi

-1

-0.5

0

0.5

1

1.5

2(b)

-pi -pi/2 0 pi/2 pi

-pi

-pi/2

0

pi/2

pi

-0.5

0

0.5

1

1.5

2(c)

-pi -pi/2 0 pi/2 pi

-pi

-pi/2

0

pi/2

pi

0

0.5

1

1.5

2(d)

-pi -pi/2 0 pi/2 pi

-pi

-pi/2

0

pi/2

pi


Beltrami Vortex Flow

• In simplified form, if the periodic vortex pattern is taken as a representative of the full spectrum, helical vortices can be described by two entangled waves with a fixed amplitude and phase relationship.

• As they intensify or weaken the amplitude and phase relationship between the vortex waves is maintained. The corresponding expansion coefficients therefore must have a very similar time-dependence.

• If this entanglement is destroyed the vortex disintegrates.


Truncation

• Spectral truncation: only the two vortex waves and one catalyst wave for each dynamical system are considered (one-triad-interactions).

• Helical truncation: for the two vortex waves only the positively helical component is considered, and for the catalyst wave only the nonhelical component.

• For the vortex waves the real part of the expansion coefficients is set identically zero, and for the catalyst wave the imaginary part.


Dynamical Systems

For both triads qualitatively the same dynamical system is obtained

with


System Equilibria

• The dynamical system has two coexisting equilibria:• The phase space origin (0,0,0) exists for all

parameter values.

exists for p 0.

• The nontrivial equilibrium

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(a)

-pi -pi/2 0 pi/2 pi-pi

-pi/2

0

pi/2

pi

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(c)


-pi/2

0

pi/2

pi

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(b)


-pi/2

0

pi/2

pi

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2(d)


-pi/2

0

pi/2

pi


Linear Stability

The linear stability of the two equilibria of each system is analysed by calculating the eigenvalues of the Jacobian matrix

evaluated at the stationary solutions.


Linear Stability

• The linear stability of the system near any of the fixed points depends on the sign of the real parts of the three eigenvalues i.

• For asymptotic stability Re[i] < 0 8 i = 1,2,3.

• Conversely, for instability Re[i] > 0 for at least one eigenvalue.

• To characterise the combined stability or instability of all three eigendirections of the linearised flow around any equilibrium, attractor and repellor strengths are defined.


Attractor Strength

The attractor strength of an equilibrium X is defined as

for all Re[i] negative, else A(X) ´ 0.


Repellor Strength

The repellor strength of equilibrium X is defined as the magnitude of the vector containing all its unstable eigenvalues

0.1

0.2

0.3

0.4

0.5

0.6

r

pq = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

r

pq

= 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.1

0.2

0.3

0.4

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

r

pq = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.1

0.2

0.3

0.4

0.5

r

pq

= 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.2

0.4

0.6

0.8

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


System States

• Without asymptotically stable equilibria there can only be transient fluctuations and the momentary development of a vortex flow would not be a convincing argument for the formation of tornadolike vortices in a turbulent fluid.

• Generally, a meaningful definition of “state of a system” requires at least some degree of persistence.

• In the following, “state” will therefore always be referring to an equilibrium of the system.


Initial and Final States

• In practice the slow velocity field is calculated from observations of the storm system prior to tornadogenesis.

• Relevant initial conditions for phase space trajectories are therefore small perturbations from the phase space origin.

• To find criteria for vortex formation it is necessary to find the background flow conditions that lead to a transition from the ground to the vortex state.


Transitions Between Equilibria

• Since stable (or unstable) manifolds of two distinct hyperbolic fixed points cannot intersect, a point arbitrarily close to the origin, which lies within the basin of attraction of the vortex fixed point, must necessarily also lie on the unstable manifold of the origin.

• Then vortex formation from small initial perturbations, i.e., a transition from a state close to the origin to the vortex equilibrium, takes place.

• To characterise the transition probability between states of the system, a combined measure of the instability of the initial state and the stability of the final state must be introduced.


Transition Probability

Based on the definition of repellor and attractor strength, the transition probability from equilibrium state X1 to equilibrium state X2 is defined as the product of the repellor strength of the initial state with the attractor strength of the final state:

0

0.1

0.2

0.3

0.4

0.5

r

pq = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.1

0.2

0.3

0.4

0.5

r

pq

= 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.1

0.2

0.3

0.4

0.5

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.1

0.2

0.3

0.4

0.5

pq

r = 1

0 0.2 0.4 0.6 0.8 10

0.25

0.5

0.75

1

0

0.05

0.1

0.15

0.2

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0

0.05

0.1

0.15

0.2

r

pq = 0.25

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1


System Parameters

• After eliminating some of the slow expansion coefficients, simple interpretations of the system parameters can be found.

• Parameters p and r determine the vertical background velocity field, and parameters q and determine the horizontal velocity field.

• Parameter q describes hyperbolic, directional shear, and parameter describes straight-line convergence and divergence.


-pi/2

0

pi/2

pi(a)


-pi/2

0

pi/2

pi(b)


-pi/2

0

pi/2

pi(c)


-pi/2

0

pi/2

pi(d)


Bifurcations

• Inside the region of stability the real parts of all three eigenvalues of the respective equilibrium must be negative.

• By continuity, at the boundaries at least one eigenvalue must have a vanishing real part.

• These transition regions in parameter space are called bifurcation points.

• There are two types of bifurcations:• Transcritical bifurcations: exchange of stability• Hopf bifurcations: creation or destruction of equilibria


Dynamical Regimes

The ground state is stable for parameter values satisfying

Since in that region arbitrary, small perturbations of the fast flow are damped out, this part of parameter space is referred to as the “laminar” region.


Dynamical Regimes

Similarly, the part of parameter space satisfying

in which the vortex state is stable, is referred to as the “tornadic” region.


Bifurcation Chart


Bifurcation Diagram


Instability Mechanisms

• In the bifurcation scenario vortex formation is described as an instability of the ground state, where the mathematical mechanism for the instability is the transcritical bifurcation.

• The transcritical bifurcation taken alone could be interpreted by saying that buoyancy or shear, or a combination thereof, must exceed a certain threshold value.

• As that forcing is increased, at some point a second critical value is reached, in which the vortex state looses stability again in a Hopf bifurcation.

• If shear instability is the primary mechanism it is conceivable that a certain minimum amount of shear is required for an initial vortex spin-up. However, a too strong background shear destroys or prevents the formation of an ordered vortex.


-pi/2

0

pi/2

pi(a)


-pi/2

0

pi/2

pi(b)


-pi/2

0

pi/2

pi(c)


-pi/2

0

pi/2

pi(d)


-pi/2

0

pi/2

pi(e)


-pi/2

0

pi/2

pi(f)


Dynamic Bifurcations

• Although the slow expansion coefficients were considered to be constant parameters for the purpose of calculating phase space trajectories, this assumption is not required in the derivation of the dynamical system.

• It is therefore possible to propose a dynamic bifurcation scenario in which the parameters are driven by the equations of motion for the slow background flow.

• For a storm flow state in the laminar region, vortical instabilities of small-scale shear zones are damped out. As the storm system evolves “into” the tornadic region vortical instabilities are amplified and reach a steady state which persists as long as the storm system remains tornadic.

• Eventually the storm evolves back into the laminar region and the vortex dissipates.


Implications for Tornado Forecasting

• Vortex formation in the simplified setting is determined by horizontal spatial variability of the embedding background flow on approximately the same scale.

• The key to understanding tornadogenesis, is therefore through understanding of the structure of these shear zones and the role of mesoscale or even synoptic-scale forcing in their formation.

• The differences between “tornadic” and “nontornadic” background flow states on the vortex scale are very subtle.


Outlook

• The effects of single triad interactions are additive for the two complex vortex waves. However, for the forcing of the catalyst waves additional linear terms have to be taken into account.

• These linear terms represent coupling between perturbation and background flow and help to specify “tornadic” or “nontornadic” background flows more completely based on the stability analysis of the vortex state.

• To be able to describe travelling vortex perturbations, it is necessary to consider both the real and imaginary parts of the fast expansion coefficients.

• The investigation of the eight-dimensional real system resulting from a combination of four complex waves with wave vectors kx, ky, kx,y, and kx,-y is planned for the near future.


Reference

Rasmussen, E. N., and D. O. Blanchard, A baseline climatology of sounding-derived supercell and tornado forecast parameters, Weather and Forecasting, 13, 1148-1164, 1998.

Return

this is the defence presentation for the dissertation dynamics of helical flow structures a...

Documents

usa slide

tornadolike vortex

storm system

formation of storm rotation

atmospheric vortex

sparse storm data

largerscale flow features

ordinary thunderstorms