Atmos. Chem. Phys., 10, 475–497, 2010
www.atmos-chem-phys.net/10/475/2010/
© Author(s) 2010. This work is distributed under
the Creative Commons Attribution 3.0 License.
Atmospheric
Chemistry
and Physics
Advective mixing in a nondivergent barotropic hurricane model
B. Rutherford1 , G. Dangelmayr1 , J. Persing1 , W. H. Schubert2 , and M. T. Montgomery3
1 Department
of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA
of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371, USA
3 Department of Meteorology, Naval Postgraduate School, Monterey, CA 93943-5114, USA
2 Department
Received: 18 June 2009 – Published in Atmos. Chem. Phys. Discuss.: 28 July 2009
Revised: 29 November 2009 – Accepted: 29 December 2009 – Published: 20 January 2010
Abstract. This paper studies Lagrangian mixing in a twodimensional barotropic model for hurricane-like vortices.
Since such flows show high shearing in the radial direction,
particle separation across shear-lines is diagnosed through a
Lagrangian field, referred to as R-field, that measures trajectory separation orthogonal to the Lagrangian velocity. The
shear-lines are identified with the level-contours of another
Lagrangian field, referred to as S-field, that measures the
average shear-strength along a trajectory. Other fields used
for model diagnostics are the Lagrangian field of finite-time
Lyapunov exponents (FTLE-field), the Eulerian Q-field, and
the angular velocity field. Because of the high shearing, the
FTLE-field is not a suitable indicator for advective mixing,
and in particular does not exhibit ridges marking the location
of finite-time stable and unstable manifolds. The FTLE-field
is similar in structure to the radial derivative of the angular velocity. In contrast, persisting ridges and valleys can
be clearly recognized in the R-field, and their propagation
speed indicates that transport across shear-lines is caused by
Rossby waves. A radial mixing rate derived from the Rfield gives a time-dependent measure of flux across the shearlines. On the other hand, a measured mixing rate across the
shear-lines, which counts trajectory crossings, confirms the
results from the R-field mixing rate, and shows high mixing
in the eyewall region after the formation of a polygonal eyewall, which continues until the vortex breaks down. The location of the R-field ridges elucidates the role of radial mixing for the interaction and breakdown of the mesovortices
shown by the model.
Correspondence to: B. Rutherford
([email protected])
1
Introduction
Several recent studies (Frank and Ritchie, 1999, 2001; Montgomery et al., 2006; Hendricks and Schubert, 2009) are devoted to the mixing of fluid from different regions of a hurricane, which is considered as a fundamental mechanism that
is intimately related to hurricane intensity. A complete understanding of these mixing processes, in particular the eyeeyewall mixing (Bryan and Rotunno 2009a, b; Cram et al.,
2007; Braun et al., 2006; Montgomery et al., 2002, 2006;
Willoughby, 2001), is expected to improve our understanding
of the physical mechanisms that regulate hurricane intensity.
Since mixing is based on particle motion, the Lagrangian
frame of reference provides the most natural framework in
which it can be diagnosed. Much progress has been made
in recent years in the study of Lagrangian mixing in twodimensional incompressible flows (Haller and Poje, 1997;
Haller and Yuan, 2000; Haller, 2000, 2001, 2002; Shadden
et al., 2005), resulting in a number of different, though related diagnostics, most of which are based on concepts from
dynamical systems theory. For applications of Lagrangian
techniques to atmospheric models, see Joseph and Legras
(2001) and Huber et al. (2001).
Much insight into specific aspects of mixing in hurricanes
can be gained from the study of simplified two-dimensional
models. Basically there are two classes of such models: Axisymmetric models, most notably the model of Rotunno and
Emanuel (1987), and planar models such as the model of
Kossin and Schubert (2001) and Schubert et al. (1999). In
this paper we apply Lagrangian techniques to analyze mixing in the planar, nondivergent barotropic model of Kossin
and Schubert (2001). Our analysis confirms a study of
Kossin and Eastin (2001) which illustrates that significant
Published by Copernicus Publications on behalf of the European Geosciences Union.
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B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
eye-eyewall mixing occurs during polygonal eyewall transitions. Lagrangian mixing in the axisymmetric model introduced in Rotunno and Emanuel (1987) is investigated in
Rutherford et al. (2009). For another discussion of axisymmetric mixing, see Wirth and Dunkerton (2006).
The model studied in this paper provides a twodimensional representation of a hurricane that initiates with
an annular ring of enhanced vorticity, and then undergoes a
vortex breakdown resulting in a monopolar end state. During the breakdown, a polygonal eyewall occurs, which forms
four elliptical pools of high vorticity. Mixing of potential
vorticity, which in this model is proportional to relative vorticity, can be visualized using Eulerian diagnostic measures
of instantaneous particle separation. A commonly used Eulerian diagnostic is the so called Q-field, derived from the
Jacobian of the Eulerian velocity field. According to the
Okubo-Weiss criterion (Schubert et al., 1999), positive values of this field indicate instantaneous particle separation,
whereas negative values indicate contraction. For our model,
the Q-field shows that regions of high relative vorticity gradient are also places where high trajectory separation and mixing occurs.
While Eulerian measures of mixing can only diagnose instantaneous particle separation, Lagrangian techniques utilize a moving frame approach along trajectories and compute measures for the average separation over a finite integration time. This approach is particularly useful in timedependent velocity fields, where trajectories may cross Eulerian streamlines (Dunkerton et al., 2009). Much of the
recent work in Lagrangian mixing has extended the ideas
of hyperbolicity for steady flows to time dependent velocity
fields (Haller, 2002; Haller and Poje, 1997; Haller and Yuan,
2000; Green et al., 2006; Ide et al., 2002), generalizing the
concept of stable and unstable manifolds of an equilibrium
to the stable and unstable manifolds of a hyperbolic trajectory. These manifolds are referred to as Lagrangian coherent
structures (LCS’s). Even in two-dimensional flows, timedependence can give rise to multiple intersections of these
manifolds, leading to a partition of the flow into invariant regions (lobes), and to mixing through the lobe dynamics (Ide
et al., 2002).
Efficient visualization of LCS’s is accomplished through
the construction of Lagrangian scalar fields, which measure separation of nearby trajectories. Current Lagrangian
methods utilize a variety of fields, including finite-time Lyapunov exponents (Haller, 2002; Haller and Poje, 1997; Haller
and Yuan, 2000), finite-size Lyapunov exponents (Koh and
Legras, 2002; Green et al., 2006), and relative dispersion
(Huber et al., 2001). Each of these methods defines a scalar
field and the LCS’s as maximal ridges of that field.
To study Lagrangian mixing in our model, we compute
particle trajectories from the numerically calculated, timevarying velocity field. The Lagrangian diagnostic fields are
functions of the initial time and position at which the trajectories are seeded. A comparison of these fields with the
Atmos. Chem. Phys., 10, 475–497, 2010
Okubo-Weiss criterion indicates that high particle separation
predicted from the Q-field typically does not coincide with
Lagrangian hyperbolic structures, however the Lagrangian
Q-field, formed by integrating Q-values along particle trajectories, shows a greater relation to other Lagrangian fields.
An important feature of the particle trajectories calculated
from our model is that they show an almost circular motion,
combined with high shearing in the radial direction. The
problem caused by this high shear is that trajectory separation and mixing occur without the entrainment of trajectories,
as the mixing is largely diffusive. A key question that we aim
to answer is whether coherent structures that play a role in the
systematic transport of trajectories can persist through high
shear.
Distinct regions of trajectories with similar properties become more difficult to distinguish through the use of scalar
fields which measure only distance, such as the field of finitetime Lyapunov exponents (FTLE-field). In fact, the FTLEfield computed from our model does not show distinguished
ridges characteristic of hyperbolic mixing. Instead, the structure of the FTLE-field is very similar to the structure of the
radial derivative of the angular velocity, indicating that the
FTLE-field is dominated by the shear and not by hyperbolic
mixing.
In order to separate shear from hyperbolic mixing we follow the approach used in Haller and Iacono (2003), and decompose the separation of trajectories in the directions along
and normal to the Lagrangian velocity. This approach allows
us to identify two Lagrangian fields: The R-field, which is
a diagnostic for hyperbolic mixing normal to the Lagrangian
velocity, and the S-field, which is a measure of shearing, and
is used to define shear-lines by its level-contours. In contrast
to the FTLE-field, the R-field shows distinct ridges and valleys observable as coherent structures. The evolution of these
structures provides a mechanism for mixing through the eyewall, and their speed indicates that this mixing is caused by
Rossby waves. The structures are particularly distinct after
polygonal eyewall formation, and they persist until the vortex breaks down, in regions where the Okubo-Weiss criterion
predicts pools of high separation associated with the formation of pools of high vorticity.
We note that another approach to diagnosing mixing in
the presence of shear is based on subtracting a mean shear
from the flow. This approach was introduced by Andrews
and McIntyre (1978) using a generalized Lagrangian mean
for nonlinear waves, and was subsequently developed further
and refined to a modified Lagrangian mean to quantify and
distinguish stirring from irreversible mixing, see McIntyre
(1980) and Dunkerton (1980).
The time-dependence of our velocity field leads to timedependent shear-lines, and regions of high orthogonal (hyperbolic) separation lead to sets of trajectories that are mixed
through the shear-lines. We quantify this mixing by introducing measured (via trajectory counting) and predicted (from
the R-field) mixing rates. In addition, we study radial mixing
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rates defined by angular averages of the FTLE-field, the Sfield, and the R-field. The mixing rates defined through the
former two fields are characteristic of shearing and give spuriously a false sense of mixing during the initial phase of the
model, where “true mixing” occurs after the polygonal eyewall formation.
In previous work on the same model, Hendricks and Schubert (2009) have applied the Lagrangian-Eulerian hybrid
method of effective diffusion (Nakamura, 1996; Shuckburgh
and Haynes, 2003). Here diffusive mixing properties are
computed based on the increasing lengths of the vorticity
contours, with the computations initialized at the initial time
of the model. The resulting mixing rate is a function of an
effective radius and the integration time, and shows similar
structures as our mixing rates.
Our methods depart from those of Hendricks and Schubert (2009) in that we utilize a moving time window, which
attributes mixing to short-time advective events. Rather than
determining contour lengths, we study transport across contours of the S-field. The resulting mixing rates are completely determined by the given velocity field, that is, they
do not depend on a chosen initial profile of the tracer distribution.
The outline of the paper is as follows. We begin, in Sect. 2,
with an overview of the nondivergent barotropic model, and
of the numerical methods used to compute the velocity field
and the particle trajectories. In Sect. 3 we introduce the scalar
fields utilized for diagnosing mixing and shear: The Eulerian
Q-field, the Lagrangian Q-field, the angular velocity field,
the Lagrangian FTLE-field, the R-field, and the S-field. The
latter two fields are extracted from the transformed variational system introduced in Haller and Iacono (2003). The
main results of the paper are presented in Sects. 4 to 6. In
Sect. 4 we study the behavior of the three Lagrangian fields
for a fixed initial time of 6 h, after a polygonal eyewall has
formed, and for different integration times. The ridge, valley,
and edge structures observed in the R-field are identified with
coherent structures and invariant sets relative to the shearing. In Sect. 5 we fix the integration time to 1 h and study
the diagnostic fields for varying initial times. The structures
observed in these fields are related to different mixing processes occurring during the three main phases of the model:
crystalization in which polygonal eyewall features form and
develop filamentation, vortex interaction and merger which
destroy the symmertry, and final collapse into a monopole.
Section 6 is devoted to the mixing rates mentioned before,
which are displayed as functions of initial time and either radius or value of S along a shear-line. Concluding remarks
and an outlook on future work are given in Sect. 7.
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2
477
Model overview
The model used in this paper is the 2-D nondivergent barotropic model for hurricane-like vortices studied
by Kossin and Schubert (2001); Kossin and Eastin
(2001); Schubert et al. (1999).
The velocity field,
u(x,t)=(u(x,t),v(x,t))∗ with x=(x,y)∗ ∈ R2 (asteriks denote transposed vectors or matrices), is given as the solution
on the f -plane of the incompressible Navier Stokes equation,
1
∂u
+ (u·∇)u − f Bu + ∇p = ν∇ 2 u,
∂t
ρ
(1)
∇·u = 0,
(2)
where
01
B=
,
−1 0
p is the pressure, ρ the constant density, f the constant Coriolis parameter, and ν the constant viscosity, chosen to be
100 m2 s−1 . In the choice of ν we follow Schubert et al.
(1999), while Kossin and Schubert (2001) used 5 m2 s−1 .
The choice of viscosity may have an effect on long time
mixing processes, which could be studied by the methods
of Hendricks and Schubert (2009). Expressing the velocity
in terms of a streamfunction ψ(x,t) as u=−B∇ψ and eliminating the pressure from Eq. (1), leads to the equation
∂ζ
∂ψ ∂ζ
∂ψ ∂ζ
+
−
= ν∇ 2 ζ,
∂t
∂x ∂y
∂y ∂x
(3)
where ζ =∇ 2 ψ is the relative vorticity. Following Kossin and
Schubert (2001), we impose periodic boundary conditions
on ψ with a fundamental domain of 600 km×600 km, and
choose as initial condition an almost circular symmetric ring
of vorticity, ζ0 (r,θ ), to model a 2-D hurricane after an initial
eyewall has formed. The defining equation of ζ0 (r,θ ) is the
equation used in Kossin and Schubert (2001).
Equation (3) was solved numerically using a Fourier
pseudospectral method with 512×512 collocation points.
Dealiasing results in 170×170 Fourier modes. The ODEsystem for the Fourier modes was solved via Matlab’s
ode45 routine, which implements a fourth order Runge-Kutta
method with adaptive time steps.
In our numerical calculation of ψ and ζ , we reproduced
the behavior observed in Schubert et al. (1999). The annular
ring of high vorticity fluid develops a wavenumber 4 asymmetry, which is present in the vorticity fields as early as 2 h,
and develops into a polygonal eyewall, with 4 mesovortices
after 6 h. After 8 h, the mesovortices begin to break down and
merge. The breakdown of the mesovortices is nearly complete after 12 h, and mixing of high and low vorticity occurs
along long filament structures. The relative vorticity fields
during these times can be seen in Figs. 6 to 11a. After 24 h,
diffusive mixing along the filaments leads to a more mixed
state. Few pools of high or low vorticity fluid remain, with a
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B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
pool of low-vorticity fluid from the eye migrating through the
eyewall, and high vorticity fluid redistributing in the eye. After 48 h, a high vorticity eye and a low vorticity environment
remain in a monopole endstate. The eyewall is no longer
present as there is no longer a strong angular velocity gradient.
In this paper we study Lagrangian mixing in the model,
which is based on following trajectories for varying initial
times. The trajectories were calculated with the same spatial
and temporal resolution as the model output, using a fourth
order Runge-Kutta method with a fixed time step of 7.5 s.
Because of time and memory limitations associated with the
large number of trajectories needed for quantifying mixing
over a sequence of initial times, the trajectories used for computing time-dependent mixing rates were calculated with a
time step of 60 s. Comparison of the results for the two time
steps for a small random set of initial conditions showed that
the use of the coarser time resolution in the mixing calculations is justified.
3
Diagnostic fields for mixing and shear
In this section we introduce the scalar fields utilized to diagnose the particle flow resulting from the numerically calculated velocity field. A main characteristic feature of the
model is an almost circular motion, the trajectories encircle the origin in the counterclockwise direction. The model
shows a strong variation of the particle speed |u| in the radial
direction. This variation leads to high shearing that dominates the particle separation, but is not the result of hyperbolic trajectory separation. Superimposed on this shear effect
is hyperbolic mixing due to trajectory separation in directions
orthogonal to the velocity.
In order to diagnose hyperbolicity, we exploit the Lagrangian field introduced in Haller and Iacono (2003), in
which hyperbolic trajectory splitting is separated from particle separation due to shearing. The more common FTLEfield is also analyzed, however, this field is dominated by the
shear and hence not suitable as an indicator for hyperbolic
mixing. In order to quantify hyperbolic mixing, we define
closed shear lines as contour lines of a suitably defined shear
field, and measure and predict transport across these lines
(Sect. 6). Further indicators used in our study are two Eulerian fields: The Hessean determinant of the streamfunction
(Q-field), and the radial gradient of the instantaneous angular
velocity.
that is, from the Hessian determinant of the streamfunction,
which is referred to as the Q-field,
2
Q(x,t) = ψxy
(x,t) − ψxx (x,t)ψyy (x,t).
According to the Okubo-Weiss criterion (Schubert et al.,
1999), regions with Q>0 show local trajectory repulsion,
whereas regions with Q<0 show local attraction. The Qfield allows diagnosis of instantaneous separation, which typically differs from Lagrangian measures of separation.
3.1.2
Angular velocity
The strong rotation and near symmetry of the flow suggests
that polar coordinates (r,θ ) provide a useful coordinate system for displaying fields calculated from the velocity field. In
particular, the quasi-circular behavior of trajectories suggests
that the angular velocity, ω=r −1 u·(−sinθ,cosθ )∗ , is an approximate measure of the particle speed, and the derivative
∂ω/∂r is an approximate measure of shearing.
For any scalar field ϕ(x,t), a measure for the radial variation is provided by the angular average, indicated by an overbar,
Z 2π
1
ϕ(r,θ,t)dθ.
ϕ(r,t)=
2π 0
Contours of ∂ω
∂r are shown in Figs. 6 to 11d–f showing the relationship of maxima (maximum normal propogating shear),
and minima (maximum counter propogating shear) of ∂ω
∂r to
features of other scalar fields.
3.2
Lagrangian fields
Let φtt0 (x0 ) be the flow map associated with the equation
ẋ = u(x,t)
(5)
for particle trajectories x(t), that is, the solution of Eq. (5)
with initial condition x(t0 )=x0 . Small perturbations in the
initial condition, y0 =x0 +ξ 0 , lead to a perturbed trajectory
y(t)=x(t)+ξ (t). For sufficiently small |ξ 0 |, the perturbation
ξ (t) can be approximated through the Jacobian of the flow
map as
ξ (t) = ∇φtt0 (x0 )ξ 0 ,
(7)
Eulerian fields
3.2.1
3.1.1
(6)
which satisfies the variational equation
ξ̇ = ∇u(x(t),t)ξ .
3.1
(4)
Finite time Lyapunov exponents
Q-field
Eulerian trajectory separation occurs when the linearized velocity shows local expansion of area. The local variation of
area can be inferred from the Jacobian of the velocity field,
Atmos. Chem. Phys., 10, 475–497, 2010
Finite time Lyapunov exponents have become a standard indicator for Lagrangian trajectory separation (Haller and Poje,
1997; Haller, 2000; Haller and Yuan, 2000; Haller, 2002).
Consider a time range [t0 ,t0 +T ] with fixed integration time
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T . The growth of |ξ (t)| during this time range is governed
by the Cauchy-Riemann deformation tensor,
∗
t +T
t +T
1(x0 ,t0 )= dx0 φt00 (x0 ) dx0 φt00 (x0 ) ,
and becomes maximal when ξ 0 is aligned with the eigenvector corresponding to the larger eigenvalue, λmax (1), of 1.
The quantity
479
fluid velocity is non-autonomous, its time variation is slow,
so that u(x(t),t) is still close to a solution of Eq. (7) for finite
times. The transformed system for η can be written in the
form (Haller and Iacono, 2003),
η̇ = A(t) + b(t)B η,
(10)
where
σ (x0 ,t0 )=
−r(x(t),t) a(x(t),t)
A(t)=
,
0
r(x(t),t)
1
ln λmax (1(x0 ,t0 ))
2|T |
is the FTLE-field (considered as a function of x0 ) at the initial time t0 . One distinguishes forward FTLE-fields (T >0)
and backward FTLE-fields (T <0). Maximal ridges of the
forward and backward FTLE-fields play the role of unstable (repelling) and stable (attracting) manifolds over the time
range considered (Haller, 2000; Shadden et al., 2005). In the
limit T →∞ or T →−∞, σ approaches the true Lyapunov
exponent, and the maximal ridges approach the stable and
unstable manifolds, of the full trajectory seeded at (x0 ,t0 ).
Data limitations in the case of numerically computed velocity fields allow only finite time integrations, and restrict the
analysis to FTLE’s.
For our velocity field, the particle separation is dominated
by the shearing in the radial direction. As a result, the FTLEvalues measure growth of perturbations in approximately angular directions, and high FTLE-values (ridges) occur near
extrema of ∂ω/∂r, whereas low FTLE-values occur near
zero contours of ∂ω/∂r. Generally, FTLE-fields are not suitable as indicators of hyperbolic mixing in the presence of
high shear.
b(t)=
3.2.2
=
Integrated Q-field
In addition to the instantaneous Q-field, Eq. (4), we consider
the integrated Q-field, formed by integrating Q along trajectories
Z t0 +T
b
Q(x,t) =
Q(x0 (t),t)dt.
(8)
1 ⊥
u ·u |x=x(t) ,
|u|2
1
(u⊥ )∗ (∇u)u⊥
|u|2
r(x,t)=
=
1
u2 vy −uv(uy +vx )+v 2 ux ,
2
|u|
and the non-diagonal entry a is composed of two parts,
a(x,t)=s(x,t)+d(x,t),
where
s(x,t)=
=
1 ∗
u (∇u)u⊥
|u|2
1
u2 uy +uv(vy −ux )−v 2 vx ,
2
|u|
d(x,t)=
1
(u⊥ )∗ (∇u)u
|u|2
1
u2 vx +uv(vy −ux )−v 2 uy .
2
|u|
The terms in the transformed linearized system (Eq. 10) motivate the definition of Lagrangian fields as diagnostics for
hyperbolic mixing and shear. Since our velocity field is
slowly varying in time, the terms associated with b(t) in
Eq. (10) are neglected in these definitions.
t0
3.2.3
Lagrangian fields for hyperbolic mixing and shear
3.3
R-field
(9)
As a consequence of incompressibility, the matrix A(t) has
the eigenvalues ±r(x(t),t). Fixing an integration time T , the
integrated field R,
Z t0 +T
t +τ
R(x0 ,t0 ) =
r(φt00 (x0 ),τ ) dτ,
(11)
are the normalized fluid velocity u/|u|, and the unit vector
orthogonal to u. This transformation is motivated by the
fact that for autonomous velocity fields, u=u(x), u(x(t)) is
a solution of Eq. (7). Although our numerically computed
describes the growth of a perturbation in the direction orthogonal to the Lagrangian velocity, and the ratio R/|T | plays
the role of a finite-time Lyapunov exponent in this direction.
Thus R is a measure of attraction (R<0) or repulsion (R>0)
of nearby trajectories towards x(t) over the integration interval [t0 ,t0 +T ]. Due to incompressibility, expansion orthogonal to u is combined with contraction in the direction of u
and vice versa.
Following Haller and Iacono (2003), in order to separate
mixing and shear in the variational system (Eq. 7), a moving frame of reference is introduced by setting
ξ = M(x(t),t)η,
where the component vectors of the matrix M,
M(x,t)=
1
(u(x,t),u⊥ (x,t)), u⊥ =(−v,u)∗ ,
|u(x,t)|
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B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
We note that in the limit T →∞, R reduces to the mixing
efficiency proposed by Ottino (1989), when this efficiency is
evaluated in the direction orthogonal to u. In our study, R
will be used as the main diagnostic field for hyperbolicity,
and in addition as a means to predict mixing rates across the
shear-lines defined below.
3.4
S-field
The term s(x,t) can be written in the form
s(x,t) = ∇|u(x,t)| ·u⊥ (x,t),
(12)
and hence characterizes the rate of change of the particle
speed in the direction orthogonal to the velocity. Thus s is
a local, Eulerian measure of shear in the fluid flow. We define the S-field by integrating s along trajectories,
Z t0 +T
t +τ
S(x0 ,t0 ) =
(13)
s(φt00 (x0 ),τ ) dτ,
t0
and use S as a Lagrangian diagnostic field for shearing.
We note that an alternative Lagrangian measure of shear
has been defined in Haller and Iacono (2003) using the nondiagonal entry of the fundamental matrix of η̇ = A(t)η.
However, this field involves a double time-integral and is
computationally more expensive. The S-field has a straightforward interpretation as shear-diagnostic due to Eq. (12),
and requires less computational effort.
As in the case of the FTLE-field, we distinguish forward
(T >0) and backward (T <0) fields for both R and S.
3.5
Shear-lines
For a given integration time T , we define the shear-line of
strength C at initial time t0 as the level contour of S, i.e.,
SC ={x0 |S(x0 ,t0 )=C}
. High values of |C| correspond to lines with high shear.
For our model, the shear lines are all closed curves around
the origin (distorted circles). Positive and negative values of
S indicate that the Lagrangian speed increases when moving radially outwards and inwards, respectively. We refer to
the first case as “normal propagating shear” and to the second case as “counter-propagating shear”. Hyperbolic mixing
measured by R is associated with transport across the shearlines. This will be used in Sect. 6 to define mixing rates.
4
Lagrangian fields and coherent structures
In Figs. 1, 2, and 3 we show forward and backward FTLE-,
S-, and R-fields, respectively, at the initial time t0 =6 h, after the polygonal eyewall has formed, and for integration
times T =±15 min, T =±30 min, and T =±120 min. For
T =±15 min, the FTLE-field (Fig. 1a,b) reveals coherent
Atmos. Chem. Phys., 10, 475–497, 2010
structures near the polygonal eyewall, since the effect of the
shear is not so pronounced over this short time range. For
increasing |T | the shear becomes dominant, and the FTLElevel contours evolve into distorted circles (Fig. 1c–f). Comparison of the azimuthally averaged fields (Sect. 5) shows
that high FTLE-values occur near extreme values of ∂ω/∂r.
While high FTLE-values correspond to high trajectory separation, they do not give clear LCS’s (ridges) for longer integration times. LCS’s can be seen only at very short integration times (Fig. 1a and b), and in regions that are predicted
by the Q-field. As integration time is increased (Fig. 1c–
f), the LCS’s do lengthen as expected, but they also become
broader. In particular, the LCS’s for T =±15 min that are located near the corners of the eyewall, converge into a single
broad ring that represents an annulus of high shear.
The S-field (Fig. 2) is a shear-indicator, and its level contours (the shear-lines) are distorted circles for all integration
times. The FTLE-field shows similar structures as the S-field
for longer integration times, confirming that trajectory separation is mainly due to shear.
The R-field (Fig. 3) shows structures of high and low Rvalues that persist over a series of integration times, making
them coherent. These structures lengthen and become more
resolved (narrower) when the integration time increases. Initial points on ridges and valleys have R>0 and R<0, indicating strong separation and contraction in the (approximately
radial) direction orthogonal to the Lagrangian velocity, respectively. The structures exist in both the forward and backward time fields, and some of the forward and backward time
structures have intersection points. Since the R-field is radially continuous, high values of the R-field lead to trajectories
that show high net movement orthogonal to the Lagrangian
velocity, and hence are more likely to cross shear-lines. Since
the structures span across the shear-lines, they are not Lagrangian, as trajectories with high angular velocity pass trajectories with lower angular velocity.
A prominent feature of the forward R-field at t0 =6 h (and
later) are the filaments observable in Fig. 3a, c and e, which
are a consequence of the polygonal eyewall. At t0 =2 h
(Fig. 4a) no filaments are observed. The filamentation concerns the ridges and valleys, as well as the edges between
them.
4.1
An advective mixing mechanism
Initial conditions x0 that satisfy R(x0 ,t0 )=0 are invariant in
the sense that there is no net movement of neighboring trajectories relative to the Lagrangian velocity. As can be seen
in Figs. 3 and 4a, ridges and valleys of R come in nearby
pairs, and the ridge and valley of a pair are separated by a
segment of a zero contour which forms (approximately) an
edge of the R-field. The edge is neutrally stable, that is, it
attracts from one side (from the ridge) and repels from the
other side (towards the valley). The situation is illustrated in
Fig. 4a ,c and d for t0 =2 h. At this initial time the motion is
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Rutherfordetetal.:
al.:Advective
Advectivemixing
mixing in
in aa nondivergent
nondivergent barotropic
barotropic hurricane
B.B.Rutherford
hurricanemodel
model
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Fig. 1. Forward and backward FTLE-fields at the initial time 𝑡0 = 6 hours integrated (a) 15 min, (b) −15 min, (c) 30 min, (d) −30 min, (e)
1201.min,
and (f)and
−120
min. FTLE-fields at the initial time t0 = 6 hours integrated (a) 15 min, (b) −15 min, (c) 30 min, (d) −30 min, (e)
Fig.
Forward
backward
120 min, and (f) −120 min.
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Atmos. Chem. Phys., 10, 475–497, 2010
8
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B. Rutherford
Rutherford et
et al.:
al.: Advective
Advective mixing
B.
mixing in
in aanondivergent
nondivergentbarotropic
barotropichurricane
hurricanemodel
model
1.6
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Fig. 2. Forward and backward 𝑆-fields at the initial time 𝑡0 = 6 hours integrated (a) 15 min, (b) −15 min, (c) 30 min, (d) −30 min, (e)
Fig.
Forward
backward
1202.min,
and (f)and
−120
min. S-fields at the initial time t0 =6 h integrated (a) 15min, (b) −15 min, (c) 30 min, (d) −30 min, (e) 120 min,
and (f) −120 min.
Atmos. Chem. Phys., 10, 475–497, 2010
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Rutherfordetetal.:
al.:Advective
Advective mixing
mixing in
in aa nondivergent
nondivergent barotropic
B.B.Rutherford
barotropic hurricane
hurricanemodel
model
9
483
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Fig. 3. Forward and backward 𝑅-fields at the initial time 𝑡0 = 6 hours integrated (a) 15 min, (b) −15 min, (c) 30 min, (d) −30 min, (e)
1203.min,
and (f)and
−120
min. R-fields at the initial time t0 =6 h integrated (a) 15 min, (b) −15 min, (c) 30 min, (d) −30 min, (e) 120 min,
Fig.
Forward
backward
and (f) −120 min.
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B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
45
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Fig. 4. (a) 𝑅-field with ridges and valleys and (b) Lagrangian speed at 𝑡 = 2 hrs. (c) Zoom of box A from 𝑅-field showing repelling edge,
Fig. 4. (a) R-field with ridges and valleys and (b) Lagrangian speed at t00 =2 h. (c) Zoom of box A from R-field showing repelling edge, and
and (d) zoom of box B showing attracting edge. Integration time is 𝑇 = 60 min. Black lines show the azimuthal velocity at the initial time.
(d) zoom of box B showing attracting edge. Integration time is T =60 min. Black lines show the azimuthal velocity at the initial time.
almost circular (Fig. 4b), but motion across shear-lines can
be observed already.
Let R be a structure (ridge or valley) of R at initial time
t0 . The structure is coherent in the sense that it evolves continuously, for varying initial time, into a structure R0 seen as
a ridge or valley of R at initial time t0 +τ . It is, however,
not Lagrangian because it is not advected with the flow, that
t +τ
is, the image of R under the flow map, Rτ =φt00 (R), has
0
advanced farther from R than R , and is not a structure of
R at t0 +τ (Fig. 5a). Generally we observe that the coherent
structures move at a slower rotational speed than that of the
Atmos. Chem. Phys., 10, 475–497, 2010
mean flow, which can be attributed to the effect of Rossby
waves (Montgomery and Lu, 1997).
A coherent structure R has a leading and a lagging end relative to counterclockwise rotation. Concerning the evolution
of R under the flow map, two cases can occur for structures
computed in a forward time integration:
(a) If the leading end is at higher angular velocity than the
t +T
lagging end, then the image RT of R under φt00
is
lengthened over the integration and tends to align with
a contour of the S-field (Fig. 5b).
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12
B. Rutherford et al.: Advective m
B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
(b) If the leading end is at lower angular velocity than the
lagging end, then the flow map rotates R, and for sufficiently large T the image RT tends to align with a
contour of the S-field in the opposite direction (Fig. 5c).
485
R
τ
R’
For a nearby pair of a ridge and a valley, the relative position of their flow map images is preserved in case a, whereas
in case b they switch position. This rotation and position
switching are a mechanism for the advective mixing during
the polygonal eyewall stage. The square eyewall gives four
valleys and ridges aligned in a way that four sets of trajectories pass from outside to inside and four from inside to
outside of an S-contour. The combination of ridge-edgevalley sets of R, aligned with the leading ends at lower angular velocity, can be seen as an indicator of fluid regions that
will roll into mesovortices over the forward integration time.
Thus the strength and size of the surrounding ridges and valleys are an indicator of the potential flux in and out of the
mesovortex.
5
Field diagnostics for varying initial time
In Figs. 6 to 11 we show in panel a the relative vorticity, in
panel b the Q-field, and in panel c the R-field with overlayed vorticity contours chosen to illustrate the relation between R-field structures, and vorticity structures. We show
the S-field, integrated Q-field, and FTLE field in panel d,
e, and f respectively, together with contours (from inside
to outside) of the maximum normal propogating shear, the
maximum tangential velocity, and the maximum counterpropogating shear. The initial times in these figures are t0 =2 h,
4 h, 6 h, 8 h, 10 h, and 12 h, and the integration time is T =1 h.
Since the shear-lines are distorted circles, we can interpret
the average σ (r0 ,t0 ) (r0 =|x0 |) as radial mixing rate. In all
b S, and FTLEFigs. 6 to 11d–f we observe that extreme Q,
b
values occur at extrema of ∂ω/∂r, demonstrating that the Q,
S, and FTLE-fields are dominated by the shear. A similar
interpretation as radial mixing rates can be attributed to the
averages S(r0 ,t0 ) and R(r0 ,t0 ). Plots of S and σ reveal these
two averages are very similar in structure, as both measure
shear. The quantity R can be interpreted as a measure of
hyperbolic mixing, which is important for transport through
the eyewall.
5.1
2–4 h: initial state
At the initial time of 2 h (Fig. 6), the model is still close
to the initial state and shows a broad ring of high vorticity
fluid. While the vorticity, Q, S, and FTLE-fields are almost
circular-symmetric, the R-field shows distinct lines of high
radial mixing, demostrating that coherent structures can persist through dominant shear. The wavenumber four asymmetry begins to show in the R-field, particularly in the forward time integration, atthough the initial vorticity profile is
nearly preserved, with any asymmetries barely noticible. The
www.atmos-chem-phys.net/10/475/2010/
R
S contour
represen
correspo
hibits th
places th
resolutio
shear.
At 8 h
is presen
similar p
the cohe
mixing.
5.3 La
(a)
R
T
S contour
R
(b)
RT
R
S contour
(c)
Fig. 5. (a) Sketch of a structure ℛ of 𝑅 at initial time 𝑡0 , the asFig. 5. (a)
Sketch ℛ
of′ aofstructure
R oftime
R at𝑡0initial
t0 , the
sociated
structure
𝑅 at initial
+ 𝜏 , time
and the
flowassocimap
𝑡00 +𝜏
ated
structure
R
of
R
at
initial
time
t
+τ
,
and
the
flow
map
image
0 non-Lagrangian nature
image ℛ
(ℛ),
illustrating
the
of
𝜏 = 𝜙𝑡 0
t +τ
Rτ coherent
=φt00 (R),
illustrating
the non-Lagrangian
of the
cothe
structures.
(b) Structure
ℛ crossing annature
𝑆 contour
with
herentabove
structures.
(b) Structure
R crossing
an below,
S contour
speed
the contour
higher than
the speed
and with
flow speed
map
above the
than the
speed
below,and
andlagging
flow map
image
image
ℛ𝑇contour
after anhigher
integration
time.
Leading
ends
of
RTand
after
animages
integration
anda lagging
R and
ℛ
their
on ℛtime.
marked by
triangle ends
and aofsquare,
𝑇 are Leading
their images (c)
on Same
RT are
a triangle
and a square,
respecrespectively.
as marked
(b) withby
opposite
orientation
of ℛ relative
tively.
(c) Sameleading
as (b) to
with
oppositeoforientation
of R relative to the
to
the contour,
a rotation
ℛ𝑇 .
contour, leading to a rotation of RT .
Atmos. Chem. Phys., 10, 475–497, 2010
The peri
intense m
state.
For 𝑡0
and the
one with
expansio
gion of
mesovor
the prim
state, No
a bifurca
ble and
here are
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during m
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S field s
Altho
distingu
14
486
B.Rutherford
Rutherford et
et al.:
al.: Advective
Advective mixing
B.
mixing in
in aa nondivergent
nondivergentbarotropic
barotropichurricane
hurricanemodel
model
−3
−7
x 10
x 10
100
100
3
2.5
2
0
1.5
−50
0.5
−100
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0
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50
−5
0
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1
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50
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100
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0
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50
100
(b)
100
100
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6
0.04
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4
y (km)
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2
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x (km)
(e)
50
100
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x (km)
50
100
(f)
Fig. 6. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆRelative
Fig.
6. (a)𝑄,
vorticity
field,
Q-field,
(c)𝑡0R-field
contours
to white
relatecontours
structures,
(d) S-field,
(e) radius
integrated
𝑄-field
and (f) 𝐹
𝑇 𝐿𝐸-field
for(b)
initial
time of
= 2 hrswith
withvorticity
integration
time 𝑇 overlayed
= 1 hr. The
in (d)-(f)
mark the
of
∂𝜔
∂𝜔
b and
Q-field
Q,
(f)
FTLE-field
for
initial
time
of
t
=2
h
with
integration
time
T
=1
h.
The
white
contours
in
(d)–(f)
mark
the
radius
of
(inner,
solid),
the
radius
of
maximum
tangential
wind
(middle,
dashed),
and
the
radius
of
maximum
counterpropogating
maximum
0
∂𝑟
∂𝑟
∂ω (inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂ω
(outer, dash-dot).
maximum
∂r
∂r
(outer, dash-dot)
.
Atmos. Chem. Phys., 10, 475–497, 2010
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B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
rotation of the R-ridges and valleys in this stage allows the
crystallization that is neccessary for mesovortex formation.
At 4 h (Fig. 7), much of the symmetry of the initial state
still remains. A forward time integration begins to display
asymmetries in the Q, FTLE- and S-fields, whereas the Rfield retains the structures that were present at 2 h. At 4 h
the backward time integration of R (not shown) also shows
regions of high orthogonal separation.
5.2
6–8 h: polygonal eyewall
At 6 h (Fig. 8), the vorticity field shows a polygonal eyewall
structure, where the flow resembles a nonlinear critical layer
for dry barotropic instability. A square inner eyewall structure forms, with pools of low vorticity fluid organized into
the corners of the eye. The fluid is largely unmixed, with
low vorticity fluid organized in the mesovortices, where it is
largely protected from the outer flow. Since vorticity is materially conserved, low vorticity fluid from the eye and environment must replace the fluid that left the eyewall. The
fluid that is mixed across the boundaries is consistent with
the ridges and valleys of the R-field. The filamentation that
develops from the stretching of high vorticity fluid that exits
from the vorticity ring to the environment, can be seen in the
form of spiral bands in the vorticity field.
The Q-field shows that, instantaneously, high trajectory
separation occurs along the square boundary of the inner eyewall. The pools of low vorticity show instantaneous contraction. The outer vorticity ring shows high trajectory separation. Even when the FTLE-fields are calculated for the
small integration time of T =3 min, there is a noticible difference between the separation points of the Q-field and the
FTLE-field. The square eyewall formation corresponds to
four structures of high FTLE-values in both the forward and
backward time fields for short integration times. As integration time is increased, the structures lengthen and are no
longer distinguishable.
The R-field shows a series of ridges and valleys that originated as coherent structures from the earlier times, but are
not as refined as previous structures. There are also structures
emanating outward from the ring of high vorticity that may
play the role of protecting the ring from interaction with the
outer flow (Dunkerton et al., 2009). If our model was a true
representation of a wave critical layer, the structures would
correspond to dividing streamlines, but the high shear prohibits this. The maximal R-regions are located at the same
places that show high FTLE-values, but with much greater
resolution than the FTLE-fields, which are blurred by the
shear.
At 8 h (Fig. 9), the polygonal eyewall structure that is
present at 6 h is still clearly visible. The R-field has similar properties as the R-field at 6 h (Fig. 8), with the coherent
structures begining to merge, showing intense mixing.
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5.3
487
Later state: mixing into a monopole
The period after t0 =8 h until t0 =12 h exhibits intense mixing
that leads to a collapse into a monopole end state.
For t0 =10 h (Fig. 10), the Q-field is less square, and the
R-field shows high mixing in two distinct regions, one with
expansion and one with contraction. The region of expansion is inside the ring of high vorticity, while the region of
contraction has become organized in the dominant mesovortex, which is the “winner” and survives to become the primary vortex during the collapse into the monopole end state.
Note that the merging of LCS’s into a single LCS is a bifurcation, and cannot happen if they represent true stable and
unstable manifolds, however the coherent structures here are
not entirely Lagrangian, yet their interactions and bifurcations play an important role in the systematic mixing during
mesovortex interaction. As more advection (stirring) in and
out of the eyewall occurs, there is filamentation of the initial
vorticity-contours with diffusive mixing occurring along the
lengthening contour boundaries, leading to an “averaging” of
vorticity values through diffusion.
For t0 =12 h (Fig. 11), the inner ring of vorticity has broken down. The FTLE- and S-fields show the outer rings
converged as a thick ring, and the model is entering the
monopole state. The R-field coherent structures now show
the dominant mesovortex migrating to the center and the
other mesovortices are disappearing due to their annihilation
by the dominant mesovortex. Regions of high R-values are
pushed outward, indicationg mixing with the outer flow. At
this stage, there is a single remaining protecting R-ridge, on
the outside of the remaining high vorticity ring, which has
served the role of protecting the mesovortex that eventually
becomes the “winner”.
Beyond 12 h, the initial regions of vorticity are not recognizable, and high (although not as high as the initial state)
vorticity fluid begins to organize into the eye. The low vorticity fluid from the eye becomes well mixed, and the eyewall
and environment become filled with relatively low vorticity
fluid. The angular velocity gradient decreases, and the Sfield shows no eyewall.
Although the fluid that is mixed beyond 12 h is not distinguishable based on its initial vorticity, the R-field still gives
regions of advective mixing, showing that the moving frame
of initial conditions still shows regions of fluid that are transported.
6
Mixing rates
The radial mixing rates σ (r0 ,t0 ) and S(r0 ,t0 ) quantify mixing due to shear, whereas R(r0 ,t0 ) quantifies hyperbolic mixing. In these mixing rates, the lines along and across which
mixing is quantified are circles. Hyperbolic mixing rates that
are more closely related to the shearing structures are mixing
Atmos. Chem. Phys., 10, 475–497, 2010
488
al.: Advective
mixing
in a nondivergent barotropic hurricane model
B. Rutherford et al.: Advective mixing inB.a Rutherford
nondivergentetbarotropic
hurricane
model
15
−3
−7
x 10
100
100
3
2.5
50
2
0
1.5
−50
0.5
−100
−50
0
x (km)
50
−5
0
−50
1
−100
0
50
y (km)
y (km)
x 10
5
−10
−100
100
−15
−100
−50
(a)
0
x (km)
50
100
(b)
100
100
0.3
6
0.2
50
50
0.1
0
y (km)
y (km)
4
0
2
0
−50
−50
0
−0.1
−100
−100
−100
−50
0
50
100
−0.2
−2
−100
−50
x (km)
(c)
0
x (km)
50
100
(d)
−5
x 10
2
100
0
−2
0
0.04
50
y (km)
50
y (km)
100
0.03
0
−4
−50
−50
0.02
−6
−100
−100
−100
−50
0
x (km)
(e)
50
100
−100
−50
0
x (km)
50
100
(f)
Fig. 7. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆRelative
𝑄-field
and (f) 𝐹
𝑇 𝐿𝐸-field
for(b)
initial
time of
= 4 hrswith
withvorticity
integration
time 𝑇 overlayed
= 1 hr. The
in (d)-(f)
mark the
of
Fig.
7. (a)𝑄,
vorticity
field,
Q-field,
(c)𝑡0R-field
contours
to white
relatecontours
structures,
(d) S-field,
(e) radius
integrated
∂𝜔
b and
(inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂𝜔
maximum
Q-field
Q,
∂𝑟 (f) FTLE-field for initial time of t0 =4 h with integration time T =1 h. The white contours in (d)–(f) mark the radius
∂𝑟 of
∂ω (inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂ω
(outer, dash-dot).
maximum
∂r
∂r
(outer, dash-dot).
Atmos. Chem. Phys., 10, 475–497, 2010
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B. 16
Rutherford et al.: Advective mixing in aB.nondivergent
hurricane
model
489
Rutherford etbarotropic
al.: Advective
mixing
in a nondivergent barotropic hurricane model
−3
−7
x 10
100
3
2.5
50
2
y (km)
y (km)
x 10
0
1.5
100
5
50
0
−5
0
−10
−50
−50
1
−15
0.5
−100
−100
−100
−50
0
x (km)
50
100
−100
−50
(a)
0
x (km)
50
100
(b)
100
100
5
0.4
4
50
50
0
3
y (km)
y (km)
0.2
0
2
0
1
−50
−50
−0.2
−100
−50
0
x (km)
50
−1
−100
−0.4
−100
0
100
−100
−50
(c)
0
x (km)
50
100
(d)
−5
x 10
100
100
1
0
50
50
0.03
−2
0
y (km)
y (km)
−1
0
−3
−50
−4
0.02
−50
−5
−100
−100
−100
−50
0
x (km)
(e)
50
100
−100
−50
0
x (km)
50
100
(f)
Fig. 8. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆRelative
𝑄-field
and (f) 𝐹
𝑇 𝐿𝐸-field
for(b)
initial
time of
= 6 hrswith
withvorticity
integration
time 𝑇 overlayed
= 1 hr. The
in (d)-(f)
mark the
of
Fig.
8. (a)𝑄,
vorticity
field,
Q-field,
(c)𝑡0R-field
contours
to white
relatecontours
structures,
(d) S-field,
(e) radius
integrated
∂𝜔
b
(inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂𝜔
maximum
Q-field Q, and
∂𝑟 (f) FTLE-field for initial time of t0 =6 h with integration time T =1 h. The white contours in (d)–(f) mark the radius
∂𝑟 of
∂ω (inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂ω
(outer, dash-dot).
maximum
∂r
∂r
(outer, dash-dot).
www.atmos-chem-phys.net/10/475/2010/
Atmos. Chem. Phys., 10, 475–497, 2010
490
Rutherford etbarotropic
al.: Advective
mixing
in a nondivergent barotropic hurricane model
B. Rutherford et al.: Advective mixing inB.a nondivergent
hurricane
model
17
−6
−3
x 10
1
x 10
100
100
3
0.5
2.5
50
50
y (km)
y (km)
0
2
0
1.5
1
−50
0
−0.5
−1
−50
0.5
−100
−1.5
−100
−100
−50
0
x (km)
50
100
−100
−50
(a)
0
x (km)
50
100
(b)
100
100
5
0.4
4
50
50
0.2
y (km)
y (km)
3
0
0
0
2
−0.2
1
−50
−50
0
−0.4
−100
−100
−50
0
x (km)
50
−1
−100
−0.6
100
−100
−50
(c)
0
x (km)
50
100
(d)
−5
x 10
1
100
0.04
100
0
50
50
−1
0
−3
−4
−50
y (km)
y (km)
0.03
−2
0
−50
0.02
−5
−100
−6
−100
−50
0
x (km)
(e)
50
100
−100
−100
−50
0
x (km)
50
100
(f)
Fig. 9. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆRelative
𝑄-field
and (f) 𝐹vorticity
𝑇 𝐿𝐸-field
for(b)
initial
time of
= 8 hrswith
withvorticity
integration
time 𝑇 overlayed
= 1 hr. The
in (d)-(f)
mark the
of
Fig.
9. (a)𝑄,
field,
Q-field,
(c)𝑡0R-field
contours
towhite
relatecontours
structures,
(d) S-field,
(e) radius
integrated
∂𝜔
b
(inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂𝜔
maximum
Q-field Q, and
∂𝑟 (f) FTLE-field for initial time of t0 =8 h with integration time T =1 h. The white contours in (d)–(f) mark the radius
∂𝑟 of
∂ω (inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂ω
(outer, dash-dot).
maximum
∂r
∂r
(outer, dash-dot).
Atmos. Chem. Phys., 10, 475–497, 2010
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B. 18
Rutherford et al.: Advective mixing in aB.nondivergent
hurricane
model
491
Rutherford etbarotropic
al.: Advective
mixing
in a nondivergent barotropic hurricane model
−7
−3
x 10
x 10
100
100
5
50
0
0
−5
3
2.5
2
y (km)
y (km)
50
0
1.5
1
−50
−50
−10
−100
−15
0.5
−100
−100
−50
0
x (km)
50
100
−100
−50
(a)
0
x (km)
50
100
(b)
0.8
100
5
100
0.6
4
0.4
50
50
3
0
y (km)
y (km)
0.2
0
2
0
1
−0.2
−50
−50
0
−0.4
−0.6
−100
−100
−50
0
x (km)
50
−1
−100
100
−100
−50
(c)
0
x (km)
50
100
(d)
−5
x 10
100
100
1
0
50
0.03
−1
−2
0
−3
−50
−4
y (km)
y (km)
50
0
0.02
−50
−5
−100
−6
−100
−50
0
x (km)
(e)
50
100
−100
−100
−50
0
x (km)
50
100
(f)
Fig. 10. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆ Relative
𝑄,
and (f) 𝐹vorticity
𝑇 𝐿𝐸-field
for(b)
initial
time of
= 10 hrs
with
integration
time 𝑇
= 1 hr. The
whitestructures,
contours in(d)
(d)-(f)
mark
radius
Fig.𝑄-field
10. (a)
field,
Q-field,
(c)𝑡0R-field
with
vorticity
contours
overlayed
to relate
S-field,
(e)the
integrated
∂𝜔
b and ∂𝜔
(inner,
solid),
the
radius
of
maximum
tangential
wind
(middle,
dashed),
and
the
radius
of
maximum
counterpropogating
of maximum
Q-field
Q,
(f)
FTLE-field
for
initial
time
of
t
=10
h
with
integration
time
T
=1
h.
The
white
contours
in
(d)–(f)
mark
the
radius
0
∂𝑟
∂𝑟 of
∂ω
∂ω
(outer, dash-dot).
maximum
(inner,
solid),
the
radius
of
maximum
tangential
wind
(middle,
dashed),
and
the
radius
of
maximum
counterpropogating
∂r
∂r
(outer, dash-dot).
www.atmos-chem-phys.net/10/475/2010/
Atmos. Chem. Phys., 10, 475–497, 2010
492
al.: Advective
mixing
in a nondivergent barotropic hurricane model
B. Rutherford et al.: Advective mixing inB.a Rutherford
nondivergentetbarotropic
hurricane
model
19
−7
−3
x 10
x 10
100
3
2.5
2
y (km)
y (km)
50
0
1.5
−50
100
5
50
0
0
−5
−10
−50
1
−15
0.5
−100
−100
−100
−50
0
x (km)
50
100
−100
−50
(a)
50
100
(b)
100
3
0.2
0
0
50
2
y (km)
0.1
−0.1
−50
4
100
0.3
50
y (km)
0
x (km)
0
1
−50
0
−100
−1
−0.2
−100
−100
−50
0
x (km)
50
100
−0.3
−100
−50
(c)
0
x (km)
50
100
(d)
−5
x 10
1
100
100
0.03
0
50
50
0
−2
y (km)
y (km)
−1
0
0.02
−3
−50
−50
−4
−100
−100
−100
−50
0
x (km)
(e)
50
100
−100
−50
0
x (km)
50
100
(f)
Fig. 11. (a) Relative vorticity field, (b) 𝑄-field, (c) 𝑅-field with vorticity contours overlayed to relate structures, (d) 𝑆-field, (e) integrated
ˆ Relative
𝑄-field
𝑄,
and (f) 𝐹vorticity
𝑇 𝐿𝐸-field
for (b)
initial
time of(c)
𝑡0 R-field
= 2 hrswith
with vorticity
integration
time 𝑇 overlayed
= 1 hr. Thetowhite
in (d)-(f)
mark the
of
Fig.
11. (a)
field,
Q-field,
contours
relatecontours
structures,
(d) S-field,
(e)radius
integrated
∂𝜔
b and
(inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂𝜔
maximum
Q-field
Q,
∂𝑟 (f) FTLE-field for initial time of t0 =12 h with integration time T =1 h. The white contours in (d)–(f) mark the radius
∂𝑟 of
∂ω (inner, solid), the radius of maximum tangential wind (middle, dashed), and the radius of maximum counterpropogating ∂ω
(outer, dash-dot).
maximum
∂r
∂r
(outer, dash-dot).
Atmos. Chem. Phys., 10, 475–497, 2010
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0.16
0.16
140
140
0.14
0.14
120
120
0.12
0.12
100
100
0.1
0.1
80
80
0.08
0.08
60
60
0.18
0.18
140
140
120
120
0.16
0.16
100
100
80
60
0.06
0.06
40
40
40
0.04
0.04
0.14
0.14
80
0.12
0.12
60
0.1
40
10
10
20
20t (hrs)
30
30
40
40
0.1
20
20
20
20
0
0
160
160
0
160
160
r (km)
r 0(km)
r (km)
r (km)
20
B.B.Rutherford
mixing
inin
a nondivergent
barotropic
hurricane
model
20B. Rutherford et al.: Advective mixing in
Rutherfordetetal.:
al.:Advective
Advective
mixing
a nondivergent
barotropic
hurricane
model
a nondivergent
barotropic
hurricane
model
493
0.02
0.02
0
0
0.08
0.08
5
10
5
t (hrs)
15
10
t (hrs)
0
20
15
20
t0 (hrs)
(a)
(a)
(a)
(a)
−3
16
14
14
12
12
10
10
8
8
6
6
4
4
2
140
140
r (km)
r (km)
120
120
100
100
80
80
60
60
40
0
120
120
100
100
80
80
60
60
40
0
−2
10
10
20
30
20t (hrs)
30
40
40
0.06
0.05
20
0.05
0.04
20
0
20
0.12
0.12
0.11
0.11
0.1
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.06
140
140
2
40
20
0
160
160
0
160
160
r (hrs)
0
r (hrs)
x 10
−3
16x 10
−2
40
t (hrs)
(b)
(b)
0
0
5
10
5
15
t0 (hrs)
10
0.04
20
15
20
t0 (hrs)
(b)
(b)
Fig. 12. (a) Averaged angular velocity 𝜔(𝑟, 𝑡), and (b) averaged
Fig. 12. (a) Averaged angular velocity ω(r,t), and (b) averaged
Fig. 12.
(a) Averaged
angular
radial
derivative
∂𝜔/∂𝑟 for
𝑡 = 0velocity
− 48 hrs.𝜔(𝑟, 𝑡), and (b) averaged
radial derivative ∂ω/∂r for t=0–48 h.
radial derivative ∂𝜔/∂𝑟 for 𝑡 = 0 − 48 hrs.
Fig. 13. Radial FTLE-values σ (r ,t ) for integration times (a)
Fig. 13. Radial FTLE-values 𝜎(𝑟00, 𝑡00) for integration times (a)
30 min and (b) 1 h.
Radial
30Fig.
min 13.
and (b)
1 hr. FTLE-values 𝜎(𝑟0 , 𝑡0 ) for integration times (a)
30 min and (b) 1 hr.
rates the
which
transport
across shear-lines
conmodel,
fieldquantify
of finite-time
Lyapunov
exponents (level
provided
tours
S
of
S,
see
Sect.
3.2.3).
C of
the
field
of particle
finite-time
Lyapunovbut
exponents
amodel,
measure
total
separation,
it did notprovided
sepaa measure
of total
particle separation,
didnot
notshow
separate
the effects
of hyperbolicity
and shear,but
andit did
6.1 Measured mixing rate
distinct
In order
to separate
thenot
effects
rate thecoherent
effects ofstructures.
hyperbolicity
and shear,
and did
show
of
the
high
shearing
occurring
in
the
model
from
hyperbolic
distinct
coherent
structures.
In
order
to
separate
the
effects
Given a level-contour SC of the S-field and an integration
mixing,
trajectory
separation
decomposed
in direcoftime
the high
occurring
inwas
the
from
Tthe
, weshearing
define the
mixing rate
Rmmodel
(C,t0 ) as
the hyperbolic
area
of initions
along
and
normal
to
the
Lagrangian
velocity.
demixing,
the trajectory
in
directial conditions
whose separation
trajectories was
crossdecomposed
SC during [tThis
,t
0 0 +T ],
composition
gave
rise
to
two
Lagrangian
fields,
the
𝑅-field
tions
along
normaloftothe
thecontour.
Lagrangian
velocity.
dedivided
byand
the length
This mixing
rateThis
is comand
the 𝑆-field,
which
thea relative
contributions
of
composition
gave
risequantified
to seeding
two Lagrangian
𝑅-field
puted
(“measured”)
by
grid offields,
initialthe
conditions
hyperbolicity
and
shearquantified
to the
mixing
process,
respectively.of
and
thecounting
𝑆-field,
which
the relative
and
trajectories
which
cross
SC . contributions
In
this approach,
andmixing
shear-strengths
identihyperbolicity
andshear-lines
shear to the
process,were
respectively.
fied
with
level-contours
and
level-values
of the 𝑆-field.
Predicted
mixing
rate
In6.2
this
approach,
shear-lines
and shear-strengths
were identified
with
level-contours
and level-values
the 𝑆-field.
The
𝑅-field
showed coherent
structuresofwhich
impacted
We define a predicted mixing rate through the R-field as foltheThe
mixing
and showed
mesovortex
interaction,
evenwhich
through
high
𝑅-field
coherent
structures
impacted
lows. Let rC be the average radius along SC . If R(x
0 ,t0 )>0
shear.
The outer
ridges of
the
𝑅-field were
also
involved
theand
mixing
and
mesovortex
interaction,
even
through
high
∗
x0 =r0 (cosθ0 ,sinθ0 ) with r0 <rC , x0 is inside the circle
in
protecting
the mesovortices
from
environmental
and
shear.
The outer
ridges of the
𝑅-field
were alsoflow,
involved
with radius rC , and the trajectory φtt0 (x0 ) is repelling. Thus
the
persistance
of theonridges
was
associated
domitrajectories
seeded
the ray
with
angle θ0 with
and the
radial
valthe
persistance
of
the
ridges
was
associated
with
the
dominance
of
particular
mesovortices
over
others.
In
contrast
to
ues slightly above r0 move outwards, towards the circle
nance
of
particular
mesovortices
over
others.
In
contrast
the
other
Lagrangian
methods
showed
that
the
𝑅-field
was
with radius rC . This suggests to define a boundary point to
able
toother
distinguish
not only
regions
of highthat
mixing,
but alsowas
the
Lagrangian
methods
showed
the 𝑅-field
(r
C −δ(θ0 ),θ0 ) through the condition
the
structures
that were
in theof
evolution
and annihiable
to distinguish
notinvolved
only regions
high mixing,
but also
lation
of
mesovortices.
Future
work
will
be
devoted
to
study
the
structures
that
were
involved
in
the
evolution
and
annihiR(r0 ,θ0 ,t0 )
elation
δ(θ
)+r
=
r
.
C
0
0
the
impact
of
coherent
structures
on
mesovortex
interaction
of mesovortices. Future work will be devoted to(14)
study
during
tropical
and its role
for the evolution
of
the impact of cyclogenesis
coherent structures
on mesovortex
interaction
on
the
θ
-ray
above
this
boundary
point
and
below
a Points
wave
critical
layer.
during tropical 0cyclogenesis and its role for the evolution of
raCThe
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be expected
to cross structures
the rC -circle.
r0 >rC proand
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the coherent
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θ
with
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0 0 quantified in terms of time- and
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cessThe
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impact of the coherent structures
on the mixing proslightly
below
r
move
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mixing
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a
cess was
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radius
r
again,
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leads
to
the
same
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point
C shear-strength as well as a function of the (avfunction
of
mixing
rates,
with
the
spatial
dependence
displayed
as
(Eq. 14), now with δ(θ0 )<0. If R(x0 ,t0 )<0, the trajectory a
erage)
radius
shear-lines. as
Overall,
the moving-frame
apfunction
of of
shear-strength
well
functionside
of the
(avis
attracting,
and
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onas
theaopposite
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proach
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paper provides
time-dependent
mixingaperage)
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Overall,
the
moving-frame
rC -circle move towards this circle, provided x0 is sufficiently
rates
that used
isolate
events
occurring
in particular mixing
time
proach
in mixing
this paper
provides
time-dependent
in protecting the mesovortices from environmental flow, and
www.atmos-chem-phys.net/10/475/2010/
rates that isolate mixing events occurring in particular time
Atmos. Chem. Phys., 10, 475–497, 2010
B.
2121
B.Rutherford
Rutherfordetetal.:
al.:Advective
Advectivemixing
mixinginina anondivergent
nondivergentbarotropic
barotropichurricane
hurricanemodel
model
494
B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
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Fig. 14.
Radial S-values
S(r
,t0 )for
forintegration
integrationtimes
times(a)
(a)3030min
min
Fig.
0 ,00𝑡, 0𝑡)
Fig.14.
14.Radial
Radial𝑆-values
𝑆-values𝑆(𝑟
𝑆(𝑟
0 ) for integration times (a) 30 min
and (b)and
1 hr.
minutes
(b)
1
hr.
minutes and (b) 1 hr.
Fig. 15. Radial R-values R(r0 ,t0 ) for integration times (a) 30 min
Fig.
𝑅-values
for
min
0 ,0𝑡,0𝑡)0 )
Fig.15.
15.Radial
𝑅-values𝑅(𝑟
𝑅(𝑟
forintegration
integrationtimes
times(a)(a)3030
min
and
(b)
1Radial
h.
minutes
minutesand
and(b)
(b)1 1hr.hr.
close to that circle. The corresponding boundary point is then
windows.
defined by (rC +δ(θ0 ),θ0 ), where δ satisfies
windows.
The
used here
,θmethods
0 ,t0 ) δ(θ ) used
TheCmethods
hereled
ledtotomixing
mixingprofiles
profilesofofsimilar
similar
eR(r
(15)
0 + rC = r0 .
structure
as
in
Hendricks
and
Schubert
(2009),
structure as in Hendricks and Schubert (2009),but
butwith
witha a
moving
time
convergence
ininLagrangian
By varying
θframe
conditions
(Eq.
14) for R>0
and
Eq. (15)
0 , theand
moving
timeframe
andwith
withfast
fast
convergence
Lagrangian
fields
integrated
over
short
times.
The
S-field
provided
for
R<0
define
an
annulus
of
initial
conditions
around
the
fields integrated over short times. The S-field provideda anatnatural
choice
of
contours
for
varying
initial
time
that
allowed
r
-circle,
whose
area
we
use
to
define
the
predicted
mixC choice of contours for varying initial time that allowed
ural
totoing
quantify
by
determining
transport
across
rate Rmixing
(C,t0 ).
mixing rate
is an approximation
pmixing
quantify
byThis
determining
transport
acrossthem.
them. of
the
measured
mixing
rate.
Of
course,
several
approximations
Future
Futurework
workwill
willaddress
addressthe
theimpact
impactofofhyperbolicity
hyperbolicityininthe
the
and simplifying
assumptions
are involved insetting,
this definition,
presence
of
high
shear
in
a
three-dimensional
presence of high shear in a three-dimensional setting,where
where
but the results
obtained make
sense and theshear
structure sepaof Rp
planar
planarshear
shearand
andmovement
movementorthogonal
orthogonaltotothe
the shearare
are sepais
similar
to
the
structure
of
R
.
m the fields introduced in this
rated
using
extensions
of
rated
usingsuitable
suitable
extensions
of the rates
fieldsσintroduced
in,tthis
Color-coded
plots
ofwill
the be
mixing
0 ,t0 ), S(r
0 0 ),
paper.
These
techniques
applied
totoa a(r
realistic,
threepaper.
These
techniques
will
be
applied
realistic,
threeR(r
,t
),
R
(C,t
),
and
R
(C,t
)
are
displayed
in
Figs.
13,
m
0 0
0 modelp that 0has both shearing and hydimensional
hurricane
dimensional
hurricane
model
that has both
shearing and
hy14,
15,
16,
and
17,
respectively,
for
integration
times
perbolic
components
governing the
mixing
processes.
The
perbolic
components
mixing
processes.
The
T =30
min
and
T =1
h.governing
Therather
rates different
σthe
and
S should
be compared
focus
of
this
work
will
be
from
that
of
a
refocus
of
this
work
will
be
rather
different
from
that
of
a
reto the average
angular Haller
velocity ω andwho
its radial derivative
cent
centpaper
paperby
bySapsis
Sapsisand
and Haller(2009),
(2009), whostudied
studiedmixing
mixing
shown in Fig. 12. These rates
are measures of shear
in∂ω/∂r
ina athree-dimensional
three-dimensionalhurricane
hurricanemodel
modelbybycomputing
computingLaLaand
show
the
highest
shear
during
the
initial
6 h, with the
grangian
grangianquantities
quantitiesalong
alonga aslow
slowmanifold.
manifold.
amount of shear dissipating as hyperbolic mixing begins to
Acknowledgements.
This
was
occur
during the polygonal
eyewall
stage.
Acknowledgements.
Thiswork
work
wassupported
supportedbybythe
theNational
National
Science
Foundation
under
NSF
Cooperative
Agreement
ATMScience
Foundationmixing
under isNSF
Cooperative
ATM,t0 ), Rm (C,t
The hyperbolic
captured
by R(r0Agreement
0 ),
0715426.
The
authors
thank
Eric
Hendricks,
Arthur
Jamshidi,
Kate
0715426.
The
authors
thank
Eric
Hendricks,
Arthur
Jamshidi,
Kate
and
Rp (C,t
).
All
of
these
rates
show
that
high
hyperbolic
0
Musgrave,
Christopher
Rozoff,
Jonathan
Vigh,
Michael
Musgrave,
Christopher
Rozoff,
Jonathan
Vigh,and
and
MichaelRiemer
Riemer
mixing
begins
with the
polygonal
eyewall
formation
at
6h
for
their
discussions.
The
authors
would
also
like
totothank
the
for
their
discussions.
The
authors
would
also
like
thank
the
and
continues
through
the
transition
to
a
high
vorticity
eye
at
anonymous
referee
and
Tim
Dunkerton
for
their
helpful
comments
anonymous
referee
and
Tim
Dunkerton
for
their
helpful
comments
24 criticisms.
h. In particular, the measured and predicted mixing rates
and
and criticisms.
R
m (C,t0 ) and Rp (C,t0 ) are very similar in structure, and reveal strong mixing near the zero S-contour S0 (jet).
References
We note that Hendricks and Schubert (2009) studied mixReferences
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6
35003500
4
4
30003000
6
6
4
4
25002500
25002500
0
20002000
0
15001500
−2 −2
2
0
0
−2 −2
10001000
−4 −4
2
15001500
C
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C
C
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C
20002000
10001000
−4 −4
500 500
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−6 −6
−6 −6
5
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10 10
15 15
20 20
5
5
t0 (hrs)
t (hrs)
10 10
15
15
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t0 (hrs)
t (hrs)
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(a) (a)
(a) (a)
35003500
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35003500
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30003000
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4
2
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25002500
2
2
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20002000
0
0
30003000
15001500
−2 −2
10001000
−4 −4
500 500
−6 −6
5
5
10 10
15 15
t0 (hrs)
t (hrs)
20 20
0
(b) (b)
C
20002000
C
C
C
25002500
15001500
−2 −2
10001000
−4 −4
500 500
−6 −6
5
5
10 10
15
t0 (hrs)
t (hrs)
15
20
20
0
(b) (b)
Fig. 17. Predicted mixing rate Rp (C,t0 ) across shear-lines SC for
Fig. 16. Measured mixing rate versus Rm (C,t0 ) across shear-lines
Fig.Fig.
16.16.
Measured mixing
raterate
versus
𝑅𝑚𝑅
(𝐶,
𝑡0 ) 𝑡across
shear-lines
Fig.integration
17. 17.
Predicted
raterate
𝑅𝑝 (𝐶,
shear-lines
𝑆𝐶 𝑆
for
versus
(𝐶,
Fig.
Predicted
𝑅𝑝 (𝐶,
shear-lines
𝑚(b)
0 ) across
𝐶 for
timesmixing
(a)mixing
30 min
and
(b)
1𝑡0h.) 𝑡across
SC forMeasured
integrationmixing
times (a)
30
min and
1 h.0 ) across shear-lines
𝑆𝐶𝑆for
integration
times
(a) (a)
30 min
minutes
andand
(b) (b)
1 hr.1 hr.
integration
times
(a) 30
minutes
andand
(b) 1
integration
times
30 min
minutes
integration
times
(a) min
30 min
minutes
(b)hr.1 hr.
𝐶 for
diffusivity yields a mixing rate that depends on r and t. Our
andand
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a high
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496
B. Rutherford et al.: Advective mixing in a nondivergent barotropic hurricane model
distinct coherent structures. In order to separate the effects
of the high shearing occurring in the model from hyperbolic
mixing, the trajectory separation was decomposed in directions along and normal to the Lagrangian velocity. This decomposition gave rise to two Lagrangian fields, the R-field
and the S-field, which quantified the relative contributions of
hyperbolicity and shear to the mixing process, respectively.
In this approach, shear-lines and shear-strengths were identified with level-contours and level-values of the S-field.
The R-field showed coherent structures which impacted
the mixing and mesovortex interaction, even through high
shear. The outer ridges of the R-field were also involved
in protecting the mesovortices from environmental flow, and
the persistance of the ridges was associated with the dominance of particular mesovortices over others. In contrast to
the other Lagrangian methods showed that the R-field was
able to distinguish not only regions of high mixing, but also
the structures that were involved in the evolution and annihilation of mesovortices. Future work will be devoted to study
the impact of coherent structures on mesovortex interaction
during tropical cyclogenesis and its role for the evolution of
a wave critical layer.
The impact of the coherent structures on the mixing process was quantified in terms of time- and space-dependent
mixing rates, with the spatial dependence displayed as a
function of shear-strength as well as a function of the (average) radius of shear-lines. Overall, the moving-frame approach used in this paper provides time-dependent mixing
rates that isolate mixing events occurring in particular time
windows.
The methods used here led to mixing profiles of similar
structure as in Hendricks and Schubert (2009), but with a
moving time frame and with fast convergence in Lagrangian
fields integrated over short times. The S-field provided a natural choice of contours for varying initial time that allowed
to quantify mixing by determining transport across them.
Future work will address the impact of hyperbolicity in the
presence of high shear in a three-dimensional setting, where
planar shear and movement orthogonal to the shear are separated using suitable extensions of the fields introduced in
this paper. These techniques will be applied to a realistic,
three-dimensional hurricane model that has both shearing
and hyperbolic components governing the mixing processes.
The focus of this work will be rather different from that of a
recent paper by Sapsis and Haller (2009), who studied mixing in a three-dimensional hurricane model by computing Lagrangian quantities along a slow manifold.
Atmos. Chem. Phys., 10, 475–497, 2010
Acknowledgements. This work was supported by the National
Science Foundation under NSF Cooperative Agreement ATM0715426. The authors thank Eric Hendricks, Arthur Jamshidi, Kate
Musgrave, Christopher Rozoff, Jonathan Vigh, and Michael Riemer
for their discussions. The authors would also like to thank the
anonymous referee and Tim Dunkerton for their helpful comments
and criticisms.
Edited by: P. Haynes
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