Jet variability in simple models
Zachary Erickson
October 2, 2014
Abstract
Zonal jets are a feature common to rotating planets, and are prevelant on earth
in the atmosphere and the Southern Ocean (SO), as well as on other planets in our
solar system. These jets are typically formed through a convergence of momentum
in regions with eddy activity. Jet formation and persistence under steady-state
conditions have been well studied, but the effects of time-varying parameters are
not well known. Here we use a simple two-layer quasi-geostrophic (QG) model to
consider how changing the friction strength over time affects the distribution of jets.
We demonstrate a hysteresis between the number of jets in the domain and the
friction in our model. A commonly used scaling for jet spacing, the Rhines length,
is not accurate for our simulations. Finally, we consider the effect of eddies on the
movement of jets in our domain through Reynolds stresses.
1
Introduction
The rotation of planets constrains fluid motion such that meridional deviations are
discouraged and zonal flow is dominant. Zonal jets are common features in planetary
atmospheres. They can be clearly seen in the banded structure of Jupiter, and are also
present in the earth’s atmosphere. Zonal motion is ubiquitous throughout all ocean
basins (Maximenko, et al., 2005), although other effects, such as the presence of continental boundaries or other external forcings, may mask this phenomenon (Rhines,
1994). In the Southern Ocean (SO), fluid motion is unconstrained by continents and
can therefore form circumpolar flows. The Antarctic Circumpolar Current (ACC) is the
dominant feature between 35–60◦ , but can also be sub-divided into various fronts which
are typically associated with individual jets (Orsi, et al., 1995; Belkin and Gordin, 1996).
Zonal jets can be maintained by a large-scale meridional temperature gradient, such
as the subtropical jets associated with the Hadley cell, or by eddies, as is the case in
the mid-latitude atmospheric jets. The latter case is the result of a convergence of
momentum into regions of eddy development, leading to an eastward flow in the stirring
region and westward flow elsewhere (Vallis, 2007).
The stability of jets is well documented (e.g. Panetta, 1993), but various observations
and models have called into question the permanence of individual jets or jet structures.
In the atmosphere, an analysis of global climate models shows a narrowing and poleward
motion of the mid-latitude jet in the southern hemisphere, which is thought to strengthen
1
Ekman transport and eddy activity in the ACC (Fyfe and Saenko, 2006). Harnik,
et al. (2014) examine an anomalous merging of two jet streams in 2010 due to an
unusually strong meridional temperature gradient; such merging events could have large
implications for weather patterns. In the ocean, the shedding of “rings” from the Agulhas
retroflection, which propagate westward and mix Indian ocean water into the Atlantic,
is the result of an instablity in the jet structure in the SO east of Africa (Lutjeharms and
van Ballegooyen, 1988). In addition, jets in the ACC exhibit instability, as they merge
and dissipate spatially and temporally throughout the region (Sokolov and Rintoul,
2007).
The formation of zonal jets is a result of conservation of potential vorticity (PV).
PV can be expressed as
∇2 ψ + f
PV =
,
(1)
h
2
2
∂
∂
where ∇2 is the Laplacian ( ∂x
2 + ∂y 2 ), ψ is the streamfunction, such that (u, v) =
(−ψy , ψx ), f is the planetary vorticity, and h is the height of the fluid layer. Conservation
of PV means a change in f can be compensated by a change in the relative vorticity
(∇2 ψ), creating a restoring force which causes zonal flow to dominate.
We choose the simplest model, to be described in Section 2, that captures the dynamics in the system. Broadly speaking, we solve the layer-wise equations
D ∇2 ψi + βy
= −δi2 f ric. − hypν,
(2)
Dt
hi
where i is the layer, D/Dt consists of the partial time derivative and advective terms
(∂/∂t + u · ∇), β is the latitudinal gradient of f , δi2 f ric. is friction present only in the
bottom layer, and hypν is the hyperviscosity. We use this model to examine how the jet
structure reacts to changes in the bottom friction.
In this formulation, friction acts as a baroclinic element in the system. Although
this model is intended mostly for oceanic applications, we do not include a wind stress
term; however, this would also act as a baroclinic feature, meaning the core dynamics
of the system can be expressed through bottom friction alone.
The rest of this paper is organized as follows. The model is introduced in Section
2, and a discussion of jet formation follows in Section 3. The variable we choose to
vary is the bottom drag (κ). Steady-state equilibrium conditions for three values of κ
are identified and characterized at the end of Section 3. In the next section, we vary κ
between these values and demonstrate a state of hysteresis between κ and the number of
jets in our domain. We then briefly discuss the role of Reynolds stresses and PV fluxes
in the movement of jets in the domain before concluding in Section 5.
2
Model
We use a two-layer, quasi-geostrophic (QG) model with a rigid lid on a beta plane
(Philips, 1954). We assume a large-scale pressure gradient (i.e. caused by a decrease
2
in averaged solar heating from the equator to the poles) which induces a large-scale
baroclinic structure in the mean zonal velocity (U ), where we define U = U1 = −U2
(subscripts refer to the upper and lower layers, respectively). PV conservation is expressed in our model as,
q1,t + J(ψ1 , q1 ) + U q1,x + (β + U λ−2 )ψ1,x = −ν∇8 q1
−2
q2,t + J(ψ2 , q2 ) − U q2,x + (β − U λ
8
(3a)
2
)ψ2,x = −ν∇ q2 − κ∇ ψ2 ,
(3b)
where
1
q1 = ∇2 ψ1 + λ−2 (ψ2 − ψ1 )
2
1 −2
2
q2 = ∇ ψ2 + λ (ψ1 − ψ2 ),
2
(4a)
(4b)
J(a, b) = (ax by − ay bx ) is the horizontal Jacobian, λ is the Rossby radius of deformation,
ν is the hyperviscosity parameter, κ is the bottom friction,
and qi and ψi are small-scale
p
perturbations. The Rossby radius λ is calculated by g 0 H/2f02 , where g 0 is the reduced
gravity, H is the average layer depth (2H is the total depth), and f0 is the average
planetary vorticity.
In our model the mean shear between the layers, 2U , is a fixed quantity, meaning
the large-scale PV gradient is constant. Eddies typically act to reduce this gradient.
Because in our model this is not possible, the fixed shear theoretically acts as an infinite
supply of energy that could be used for jet or eddy production. In our simulations,
the damping effect of bottom friction (along with hyperviscosity) allows our model to
equilibrate.
We avoid the influence of boundaries in x and y by making our model doubly periodic.
Longitudinal periodicity is justified because the jets we are interested in modeling are
circumpolar. Latitudinal periodicity stems from the idea that there exists a large-scale
global gradient in pressure, and we are interested in modeling a small portion in the
middle of this gradient, far from any boundary effects. While this greatly simplifies
the system, lack of boundaries does not mean that the domain size is unimportant.
Specifically, L/Ljet , where L is the lenth of the (square) domain and Ljet is the average
jet spacing, is constrained to the set of integers because there cannot exist a fractional
jet. In our simulations, L/Ljet ∼ O(10). This “rounding” effect means that a change in
a system from n to n ± 1 jets may be trivial.
3
3.1
Jet formation and characteristics
Jet formation
We start each simulation with negligible q1 and q2 . The nonlinear terms in (3) are
therefore small and can be ignored. If we neglect both dissipative effects, our system of
3
Figure 1: a: Contours of growth rate s during the linear stage as a function of zonal
and meridional wavenumbers k and l, calculated at κλ/U = 0 and βλ2 /U = 0.02. b:
Contours of maximum growth rate s during the linear stage as a function of β and κ,
calculated at k = λ and l = 0.
equations becomes
q1,t + U q1,x + (β + U λ−2 )ψ1,x = 0
−2
q2,t − U q2,x + (β − U λ
)ψ2,x = 0.
(5a)
(5b)
Using the ansatz (ψi = ψ̂i ei(kx+ly−ωt) ), we find that the growth rate s = Im(ω) is at a
maximum for (k, l) ∼ (0, λ), where k and l are the zonal and meridional wavenumbers,
respectively (Figure 1a). Shear in U between layers acts as a source of energy to waves
of all k and l, and the waves which have the maximum growth rate dominate. This stage
is generally known as the “elevator mode”, since k = 0 means it has no structure in the
meridional direction (Figure 2a).
Friction acts to reduce the total amount of kinetic energy in the system. Interestingly,
this does not mean an increase in κ always decreases the amount of eddy kinetic energy
in the initial growth stage. If we include the frictional term in our analysis, we find
that, for most values of β, the maximum growth rate, which stays relatively constant at
(k, l) ∼ (0, λ) decreases as κ increases (Figure 1b). However, when β becomes large, an
increased bottom drag actually increases the growth rate, presumably because friction
constrained to the lower layer increases the baroclinic component of the system, which
can act as a source of eddy kinetic energy (Thompson and Young, 2007).
4
Figure 2: a: Contours of q1 (upper layer) during the linear growth stage, also known as
the elevator mode (tU/λ = 30). b: Beginning of the transition state, where non-linear
terms become important (tU/λ = 40). Note the change in color scale. c: Sample q1 field
in steady state (tU/λ = 200). q has a zonal structure, but many eddies are also present.
The magnitude of q1 is somewhat less than during the transition state. All snapshots
are taken from a simulation with βλ2 /U = 0.5, κλ/U = 0.1, and ν/(U λ7 ) = 0.25.
An increase in β monotonically decreases the maximum growth rate. Low β means
there is only a small meridional change in the planetary vorticity. Therefore, fluid
parcels can be displaced in the y direction with little restoring force. High β, conversely,
designates a large change in the planetary rotation with a change in latitude, which acts
to more heavily constrain flow to the x direction.
As the elevator mode grows, q increases and the non-linear terms become important.
There is a short transition state, in which the eddy kinetic energy reaches its maximum,
and zonal motions become dominant. After time, the model reaches a steady-state
equilibrium at (k, l) = (kjet , 0), which describes zonal jets (Figure 2b,c).
Solving a priori for the jet spacing, Ljet , is non-trivial. One common scaling is
known as the Rhines length (Rhines, 1975). For a single layer over flat bathymetry, (1)
becomes
P V = ∇2 ψ + βy
(6)
where the second and third terms are the advective and planetary terms and dissipative
effects are neglected. From this we can determine the Rhines length scale
s
Ueddy
LRh ∝
,
(7)
β
where Ueddy is typically taken as the root-mean-squared eddy velocity. Assuming the
2
proportionality constant is O(1), LUβ 1 implies that the beta-term dominates and
the non-linear advective terms are negligible, leading to wave-like, sinusoidal behavior.
2
Conversely, when LUβ 1, the non-linear advective term dominates. The beta term
5
restricts meridional motion, and the advective term, through the strength of the eddies,
mixes fluid meridionally. At some point (the proportionality constant) these effects
cancel each other out, giving the length scale associated with jets.
3.2
PV staircase
The total PV, Q, in the top layer is given by
Q1 = q1 + (β + U λ−2 )y.
(8)
Initially, q1 is small and Q1 is essentially a linear function of y. The elevator stage
contains only small perturbations in q1 , but as jets form |q1 | increases. Eastward jets are
bounded by cyclonic eddies (q > 0) on the north flank and anti-cyclonic eddies (q < 0)
to the south. Westward jets are opposite, leading to westward jets forming in areas of
PV homogenization and eastward jets being associated with large latitudinal gradients
of PV (Dritschel and McIntyre, 2008).
3.3
Equilibrium states
The important three parameters we control are the planetary gradient in vorticity
(β), the bottom drag (κ), and the hyperviscosity (ν). Non-dimensionalized, these are
2
κλ
ν
∗
∗
expressed as β ∗ = βλ
U , κ = U , and ν = U λ7 . We expect the hyperviscous parameter
to have a negligible influence on the dynamics of the system (Panetta, 1993; Thompson
and Young, 2006). Correct selection of ν is model-based and dependent on the model
resolution. This parameter is responsible for small-scale dissipation in the system. As
the resolution increases ν can be decreased because finer-scale structures are possible.
Our model uses 256 × 256 resolution, and we empirically find ν ∗ = 0.25 to work well.
A small number of tests at 512 resolution and the same ν ∗ did not give qualitatively
different results.
Thompson and Young (2007) recently used a similar model to study the effects of
β ∗ and κ∗ on the eddy diffusivity of temperature, Dτ . This diffusivity is related to
the eddy strength, which we expect to affect jet spacing through (7). They found that
the dependence of Dτ on κ varied strongly with respect to β. For β ∗ ∼ 0.7, Dτ was
essentially independent of κ, and the dependence increased as β ∗ moved away from 0.7.
We tested a variety of conditions before finding a suitable value of β ∗ such that an
adjustment in κ∗ would effect a large change in jet spacing. Our chosen equilibrated
states, with β ∗ = 0.5 and κ∗ = [0.1, 0.02, 0.005], are shown in Figure 3.
4
4.1
Results and discussion
Hysteresis
We start with the equilibrated states in Figure 3 and vary the bottom drag within
= [0.1, 0.02, 0.005] to evaluate how the jets react. One such simulation is shown in
Figure 4, where we initialize the jet structure at κ∗ = 0.1 and cycle stepwise through our
κ∗
6
Figure 3: Snapshot of zonal jet structure in our three systems, with β ∗ = 0.5, ν ∗ = 0.25,
and κ∗ = a: 0.1, b: 0.02, and c: 0.005 after they have equilibrated (tλ/U = 2000).
frictional parameter space. Similar models have shown that jets are temporally quite
stable (Panetta, 1993). Our simulation shows that jets also have considerable stability
to changes in friction, remaining in the same structure during a 5-fold decrease in κ∗
from 0.1 to 0.02. However, when the drag is further decreased to κ∗ = 0.005 the jet
structure “breaks” and the jets reform into a new equilibrium. Similarly, as the drag
is increased, the initial step only strengthens the jets, and another step is required to
re-form them.
Friction acts to oppose fluid motion. We might expect that a lower frictional term
would increase the energy available to the eddies, causing them to become stronger and
increasing the jet spacing, as shown in (7) and Figure 3. We define
q
ui,jet = ui ,2
(9a)
q
ueddy = (u1 − u1,jet )2 + v12 + (u2 − u2,jet )2 + v22 ,
(9b)
where the overbar denotes a horizontal average (in x and y). These values are shown
in Figure 4c,d. The first step-change in bottom drag initially increases ueddy , but that
extra energy is quickly taken up by ujet , and the jets intensify. Initially, the second drop
in drag follows the same formula, as ujet increases once more. However, as if a threshold
value was crossed, some of the energy available stays in ueddy , which increases, peaking
as the jet structure begins to break up at at tλ/U = 4250.
As the jets reform at a larger spacing, eddy energy decreases, which is counterintuitive from the Rhines scale. Weaker jet activity evidently has an effect on eddy
strength. The low drag may also have decreased the baroclinicity of the system enough
to decrease the eddy kinetic energy from that component. We see that as the friction
increases again at tλ/U = 6000 both ujet and ueddy increase, even though the jet structure does not change. It is only when κ∗ is increased back to 0.1 that the jets again
re-form themselves.
7
Figure 4: a: Hovmoller of non-dimensionalized ux over time. Drag changes stepwise as
indicated in b. ueddy and ujet (for the upper layer) are shown in c and d.
A step-change in drag is not very physical. Many of the same experiments were performed with a slow, linear change in κ, which did not qualitatively change the observed
effects. In one simulation, shown in Figure 5, we initialized the system at κ∗ = 0.1 for
2000 model days, slowly decreased κ∗ to 0.005 over 3000 model days, and held it there
for an additional 2000 model days. The jets kept their position until tλ/U ∼ 5100,
which was after κ∗ had leveled out at 0.005. We find that while the peak in ueddy also
occurs at tλ/U ∼ 5100, the increase in eddy velocity begins at about tλ/U = 4630,
when κ∗ ∼ 0.016. Throughout the decrease in drag, and indeed until the jet structure
breaks up, ujet is increasing, meaning it is constantly taking excess energy out of ueddy .
However, there appears to be a tipping point at κ∗ 0.016 where ujet is no longer able
to take up all of this energy. ueddy grows, reaching its maximum as the jets reorganize
themselves, implying that the change in eddy strength caused the jets to reform — as
expected by (7).
4.2
Reynolds stresses and PV fluxes
We now consider how jets reorganize. A jet can be considered as an accumulation of
zonal momentum in the mean flow. The zonal energy equation in the upper layer is
D
KE 1 = −u1 (u01 v10 )y ,
(10)
Dt
where the overbar denotes a zonal and time average. The term on the right hand side
involves the divergence of the Reynolds stress, u0 v 0 . Reynolds stresses arise from the
8
Figure 5: a: Hovmoller diagram of non-dimensionalized ux over time for a simulation
which linearly decreases κ∗ from 0.1 to 0.005, as shown in b. Also shown is ueddy and ujet
over time (c,d). Note that the large spike in ueddy begins as the drag is still decreasing
but the peak, which corresponds to the break-up in the jet structure, occurs after the
drag has stabilized.
interplay between eddies and jets. An isotropic eddy has u0 v 0 = 0. When eddies are
sheared by an eastward mean flow, eddies on the southern flank of the jet develop positive
Reynolds stresses, whereas u0 v 0 on the northern flank is negative. Thus, (u0 v 0 )y < 0 and
energy flows from the eddies into the mean flow.
The divergence of the Reynolds stress can also be related to the meridional flux of
PV. In the upper layer, friction is non-existent and, if we neglect hyperviscosity,
q1,t + J(ψ1 , q1 ) = 0.
(11)
Taking the zonal and time mean, we find that, in a domain with x periodicity,
(v10 q10 )y = −q1 ,t .
(12)
Thus, in steady state the eddy meridional flux of PV is constant. From (4a), it can be
shown that
1
(13)
(u01 v10 )y = −v10 q10 − λ−2 ψ1,x ψ2 .
2
This is a variation on the Taylor-Bretherton identity. In the lower layer, we must account
for friction. Neglecting the hyperviscous term,
q2,t + J(ψ2 , q2 ) = −κ∇ψ2 .
9
(14)
We take the zonal and time mean, and recover
(v20 q20 ) = −q2,t + κu2,y .
(15)
The eddy meridional flux of PV in the lower layer is therefore directly related to the
mean flow. Jets are largely barotropic, meaning u1 ∼ u2 (this is less true in the cores of
the jets). The meridional flux of PV in the lower layer could thus act as a transfer of
energy between the layers. We can more clearly show this transfer by using (4b) to find
1
(u02 v20 )y = −v20 q20 + λ−2 ψ1,x ψ2 .
2
(16)
The term ψ1,x ψ2 acts to transfer energy between the layers. Combining (13) and (16),
v10 q10 + (u01 v10 )y + v20 q20 + (u02 v20 )y = 0,
(17)
where in steady state v10 q10 is a constant, v20 q20 = f (u2 ), and (u01 v10 )y and (u02 v20 )y act
directy on u1 and u2 , respectively (although the Reynolds stresses in the lower layer are
typically quite small).
Figure 6 shows the relationship between non-dimensionalized u1 and −u1 (u01 v10 )y
for a stationary jet and a jet which is moving northward. For the stationary jet, the
Reynolds stress divergence term is at a maximum at the center of the jet. In this case,
the eddies serve to strengthen and narrow the jet. The moving jet, however, has a
Reynolds stress divergence which is slightly offset in the direction of motion of the jet,
showing that Reynolds stresses play a role in moving the jet. In addition, the Reynolds
stress divergence term is negative on the southern flank of the migrating jet, meaning
here energy is being taken out of the mean flow and put into the eddies.
5
Conclusion and future work
We used a simple quasi-geostrophic model to demonstrate an inherent hysteresis
between jet spacing and bottom friction. The interaction between eddies and the mean
flow is highly significant. Decreasing bottom drag increases the strength of eddies, which
can eventually become strong enough to force jets to migrate and merge. Eddies can
themselves move jets through the meridional gradient of Reynolds stresses, according to
(10). We find a connection between the meridional flux of PV and the divergence of the
Reynolds stresses in both layers in (17). Using simplifications discussed in the previous
section, in steady-state we can re-write (17) as
(u01 v10 )y ∼ −κu2 + C,
(18)
where C is a constant composed of v10 q10 and the constant of integration involved in
solving (15) and we neglect any contribution by (u02 v20 )y as small. Note that when
qi,t 6= 0, additional complexity enters through (12) and (15).
We have found occasions in our simulations where threshold values of κ seem to have
been reached which cause the jets to reform. We see these in the evolution of ueddy ,
10
Figure 6: Blue lines show the non-dimensionalized zonal jet u/U ; green lines show
the contribution to the mean flow from the non-dimensionalized eddy component,
−u(u0 v 0 )y Ljet /U 3 /10. Dashed lines are values averaged over the zonal domain size and
500 time units during which the κ∗ is constant and the jets are in steady state. Solid
lines are values averaged over the zonal domain size and 20 time units during with κ∗ is
linearly increasing from 0.02 to 0.1 and the chosen jet is migrating northward (upward).
The y axis is set so that y = 0 is the center of each jet, meaning y = 0 changes position
in the domain as the jet moves.
which does not much vary with changing drag outside of these threshold conditions.
Future work involves a careful study of these values of κ∗ which seem to greatly affect
the system.
Little time has been spent analyzing the second half of Figure 4, where the drag
increases. A curious effect between eddy strength and jets was observed, where a small
number of jets in the domain coincides with low ueddy , in defiance of the Rhines scaling (7). One way forward may be through (18). As κ decreases, the divergence of the
Reynolds stress in the upper layer decreases as well, meaning there is less transfer between the jets and the eddies according to (10). In addition, a lower u2 also decreases
this transfer term. Jets are largely barotropic, so small u2 implies small u1 and a weak
jet, which is what we see during the κ∗ = 0.005 stage of Figure 4. A better understanding
of how eddies and jets feed on and off each other is therefore a future goal.
Finally, it is unclear to what extent conclusions derived from this highly simplified
model relate to the real world. We plan to use satellite data from AVISO (Archiving,
Validation, and Interpretation of Satellite Oceanographic Data) to detect zonal jets and
upper layer Reynolds stress divergences in the ocean, especially where jets are strong
and variable, such as leeward of the Kerguelen plateau in the SO and in the Kuroshio
extension in the northwest Pacific. AVISO uses satellite altimetry to generate temporal
maps of sea surface height (SSH), from which geostrophic flow can be determined. From
this flow the jet structure and Reynolds stresses in the upper layer can be approximated,
and we can investigate to what extent (10) holds.
11
Acknowledgements
ZE would like to thank Prof. Andrew Thompson for his guidance in this project, as
well as the Thompson group for discussions related to this topic.
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