A model for eastward and westward jets in laboratory experiments and planetary atmospheres

Marcus, P. S.; Lee, C.

doi:10.1063/1.869668

Flows in a rotating annular tank [J. Sommeria, S. D. Meyers, and H. L. Swinney, Nonlinear Topics in Ocean Physics, edited by A. Osborne (North Holland, Amsterdam, 1991); Nature (London) 337, 58 (1989); T. H. Solomon, W. J. Holloway, and H. L. Swinney, Phys. Fluids A 5, 1971 (1993); J. Sommeria, S. D. Meyers, and H. L. Swinney, Nature (London) 331, 689 (1989)] with a sloping bottom (that simulates a barotropic atmosphere’s Coriolis force with a topographic $β$ -effect [J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed. (Springer, Berlin, 1986)]) produce eastward and westward jets, i.e., azimuthal flows moving in the same or opposite direction as the annulus’ rotation. Flows are forced by pumping fluid in and out of two concentric slits in the bottom boundary, and the direction of the jets depends on the direction of the pumping. The eastward and westward jets differ, with the former narrow, strong, and wavy. The jets of Jupiter and Saturn have the same east–west asymmetry [P. S. Marcus, Ann. Rev. Astron. Astro. 431, 523 (1993)]. Numerical simulations show that the azimuthally-averaged flow differs substantially from the non-averaged flow which has sharp gradients in the potential vorticity $q$ ⁠. They also show that the maxima of the eastward jets and Rossby waves are located where the gradients of $q$ are large, and the maxima of the westward jets and vortex chains are located where they are weak. As the forcing is increased the drift velocities of the two chains of vortices of the eastward jet lock together; whereas the two chains of the westward jet do not. Inspired by a previously published, [P. S. Marcus, Ann. Rev. Astron. Astro. 431, 523 (1993)] piece-wise constant- $q$ model of the Jovian jets and based on numerical simulations, a new model of the experimental flow that is characterized by regions of undisturbed flow and bands of nearly uniform $q$ separated by sharp gradients is presented. It explains the asymmetry of the laboratory jets and quantitatively describes all of the wave and vortex behavior in the experiments including the locking of the vortex chains of the eastward jet. The simulations and new model contradict the predictions of a competing, older model of the laboratory flow that is based on a Bickley jet; this raises concerns about previous calculations of Lagrangian mixing in the laboratory experiments that used the Bickley model for the fluid velocity. The new model, simulations and laboratory experiments all show that jets can be formed by the mixing and homogenization of $q$ ⁠. The relevance of this to the jets of Jupiter is discussed.

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f

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U

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24.

For the westward jet model (in an unbounded domain)

χ

is given by:

{χ(1+k}_{0} {)−2=[χ(1−k}_{0} {)+2]ηe}^{−2k(a+b)}

where

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and

k_{0} = \sqrt{1−2/[kχ(a+b)]}

⁠. The upper (lower) sign in the definition of

η

is for eigenmodes that are symmetric (anti-symmetric) about

y=0

⁠. For eastward jets, the eigenmodes are always symmetric; otherwise, there is no disturbance of the contours of constant

q

at

y=0

which is contrary to the laboratory and the numerically calculated flows, but for westward jets the eigenmodes could be antisymmetric, although in the initial-value calculations and experiments they are always symmetric. For

Y_{2}^{slit} =a

⁠,

η

is

+1

for symmetric and

−1

for anti-symmetric modes. Tables III and IV are computed with symmetric modes. For eastward jets with (with one or two discontinuities)

χ

approximately satisfies

{χ(k}_{0} {+1)−2=[χ(k}_{0} {−1)+2]e}^{{−kk}_{0} {(|Y}_{1}^{slit} {−Y}_{2}^{slit} |−2a)}

with fractional errors of

{O(e}^{−2k(a+b)} {/k}^{2} {(a+b)}^{2})

⁠. If there are boundaries within a distance

L_{b}

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χ

of order

e^{{−2kL}_{b}}

⁠. Under most circumstances the wavenumbers of interest are large and the discontinuities are far from the boundaries, so

k(a+b)

and

{kL}_{b}

are large, the fractional errors are small, and the boundary locations do not usually affect

c

⁠. For eastward jets, the fractional errors in

{[ū(y}_{d})−c̄]

in equation (27) contain the errors inherent in

χ

from above as well as errors of

{O(e}^{{−2kk}_{0} {(Y}_{2}^{slit} −a)})

⁠. For the westward jets, equation (27) has fractional errors of

{O(e}^{−2k(a+b)} {/k}^{2} {(a+b)}^{2} {,e}^{{−2kL}_{b}})

for

{2k(Y}_{2}^{slit} −a)≫1

and

{O(e}^{−2k(a+b)} {,e}^{{−2kL}_{b}})

otherwise.

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Our model flows also have Rossby waves that are not localized to the locations where there are jumps in

q̄

⁠. These eigenmodes are sines and cosines in

y

rather than exponentials. They exist in regions where

ū

is constant and are similar to the usual Rossby waves in a channel with

{c̄≃ū−β/[k}^{2} {+(nπD)}^{2}]

where

n

is an integer and

D

is the width of the region of constant

ū

⁠. Since there is no forcing in the regions of constant

ū

⁠, there would be no reason for these modes to be excited. Indeed, our numerical calculations show no evidence of these channel Rossby waves, so we do not consider them.

27.

{Δq}_{2}

and

c_{2}

are further reduced by the proximity of

R_{out}

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R_{2}^{slit} .

Note that

{Δq}_{2}

is equal to

{β(a}_{2} {+b}_{2})

⁠. The value of

b_{2}

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{(R}_{2}^{slit} {+b}_{2})

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28.

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29.

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q

is much closer to

R_{out}

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30.

C. Lee and P. S. Marcus, “Eastward and westward jets in a rotating annulus model of a barotropic atmosphere,” in the Proceedings of the CERCA Workshop in Semi-analytic Methods in Turbulence, edited by K. Coughlin (Montreal, Canada, 1997).

31.

The large uncertainties in the averaged jovian velocities make it necessary to infer

q

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q

of the east–west velocity approximately equal to a step-function. Use of the traditional sinusoidal east–west velocity was particularly bad. When the initial

q

was approximately a step function, the chains of cyclonic and anti-cyclonic vortices were intertwined and centered at the westward jets, similar to the observed jovian vortex streets. An initially sinusoidal east–west velocity produced vortex chains widely spaced in latitude with each chain located near a local zero of the east–west velocity. Inclusion of a finite Rossby deformation radius in the QG equations or use of the shallow-water equations made these results even more pronounced.

32.

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ū

is a series of connected

\cosh

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40.

Computations with parabolic, Gaussian, and top-hat

w (r)

with the same

L^{slit}

give the same

P_{crit}

to within 1%, where

L^{slit}

is the full width at half maximum of

w (r)

⁠. Figures 13 and 14 used a parabolic

w (r)

⁠. In principle, one could use the experimental values of

P_{crit}

with fig. 14 to determine the best value of

L^{slit}

to use in numerical computations. However, this method cannot be used at

{P>P}_{crit}

because the “best” value of

L^{slit}

depends on

P

⁠:

L^{slit}

controls the vortex layer’s thickness. As

L^{slit}

increases, the

ω(r)

at each slit goes from a delta function to a function with width approximately equal to

L^{slit}

⁠. With

ν→0

equation (5) shows that

u

of the primary flow is approximately independent of

L^{slit}

⁠, so at each layer

q≃ \frac{∓fτP}{{2πHR}_{i}^{slit} L^{slit}} {+βR}_{i}^{slit} .

As the layers roll-up into discrete vortices, the

q

of each vortex is approximately conserved at its initial value. Thus,

L^{slit}

determines the strength of the vortices in the chains. With

ν→0

equation (5) also shows that the potential circulation of the layers and chains are nearly independent of

L^{slit}

⁠; therefore, the areas of the vortices must depend on

L^{slit}

⁠.

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