Environ Fluid MechDOI 10.1007/s10652-014-9349-0

ORIGINAL ARTICLE

Instabilities in non-rotating and rotating shallow shearflows

Vincent H. Chu

Received: 11 February 2013 / Accepted: 26 February 2014© Springer Science+Business Media Dordrecht 2014

Abstract Numerical simulations for the wave radiation effect on the linear and nonlinearinstabilities of rotating and non-rotating shallow flows are conducted using shallow-waterequations. At a low convective Froude number, the results of the instabilities is a string ofeddies. The coalescence between the neighbouring eddies decides the transverse mixing ofthe shallow shear flow. At a higher convective Froude number, the development of the shearflow is characterized by wave radiation and the production of shocklets. The radiation ofwaves in the non-rotating shallow flow is a phenomenon analogous to the radiation of soundin gas dynamics. In the rotating flow on the other hand, the shallow-flow instabilities areintensified due to rotational interference within a window of instability over a narrow rangeof Rossby numbers.

Keywords Shear instability · Wave radiation · Rotational interference · Eddies · Shocklets ·Shallow flow · Numerical simulation

1 Introduction

Shear instabilities has been studied for body-force and wave radiation effects on mixingand turbulence [5,9,22]. In shallow waters, bottom friction has been identified as a body-force effect. Much of the early research into the bottom-friction effect on transverse shearinstability in shallow waters was completed by Professor Gerhard Jirka and his colleagues[11–13,17,35,37,40]. However, these works were formulated for shallow flow using a rigid-lid approximation for flow with relatively low Froude number [1,2,10,35]. More recentstudies have shown the dependence of the transverse shear instability on Froude number[15,30,31]. In open-channel flow, the dependence is associated with surface gravity waves. Ingravity current, the internal waves are responsible for the radiation of energy. The densimetric

V. H. Chu (B)Department of Civil Engineering and Applied Mechanics, McGill University Montreal,Montreal, QC, Canadae-mail: Vincent.chu@mcgill.ca

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Froude number can be quite large, leading to depth and velocity discontinuities and theformation of shock waves in the gravity current [7]. Recognizing the importance of shockwaves, Jones et al. [18] has developed shock-capturing numerical scheme for simulation ofthe gravity current.

Most previous studies on the instability of shear flow were based on linear stability analy-sis (LSA), treating the phenomenon as an eigenvalue problem of linearized shallow-waterequations. In the present investigation, numerical simulations were conducted for shear insta-bilities using fully nonlinear shallow-water equations. The fully nonlinear simulation (FNS)determines the shear instabilities as surface waves are radiating away from the transverseshear flow in shallow waters, and as internal waves are radiating away from gravity current.The radiation of surface and internal waves are phenomenon analogous to the radiation ofsound in gas dynamics. The present series of FNS has reproduced the results of LSA fornon-rotating shear flow and, most significantly, has produced the nonlinear development ofthe instabilities in both rotating and non-rotating shear flow in shallow waters.

This paper has essentially two parts. The effect of wave radiation on shear instability isstudied first for the non-rotating flow. Next, we examine a simulation of the Coriolis effectdue to the Earth’s rotation. This produced an unexpected phenomenon, as the rotation wasfound to interfere with the wave radiation. The rotation suppressed the wave radiation, leadingto the excitation of the shear flows in a window of instability centered on a certain criticalRossby number.

2 Simulations using shallow water equations

The dynamics of transverse shear instabilities in shallow waters are simulated numericallyusing shallow-water equations (SWEs) [8,14]:

∂h

∂t+ ∂uh

∂x+ ∂vh

∂y= 0 (1)

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y− f v = −g

∂h

∂x(2)

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y+ f u = −g

∂h

∂y(3)

where g = gravity, f = Coriolis parameter, h = depth, and (u, v) = velocity. Friction forceis ignored in the present formulation.1 The SWEs are depth-averaged equations for open-channel flow. They are also one-layer model equations for gravity currents. For one-layermodel of density difference � ρ, the symbol g in the SWEs is the gravity reduced by a factor(� ρ / ρ). The numerical simulation begins with the introduction of a small perturbation(u′, v′, h′) to the parallel base flow [U(y), 0, H(y)]:

u = U (y) + u′, v = v′, h = H(y) + h′ (4)

The subsequent development of (u′, v′, h′) are calculated in a staggered grid using a trans-critical shock-capture scheme developed by Pinilla et al. [32] and Wang et al. [38,39]. Fourth-order Runge–Kutta time integration is part of the scheme. The mesh is uniform in bothdirections containing 200 finite volumes over one wave length. For linear instabilities, thedomain of computation is one wave length in the longitudinal x-direction and at least four

1 The friction effect can be significant [10,11]. It is ignored in the present formulation so that attention isfocus on wave radiation and rotating effects.

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wave lengths in the transverse y-direction; six wave lengths in the y-direction are used forFroude numbers greater than 0.7. No detectable difference in results when the simulationswere conducted using the coarser grid of 100 finite volume over one wave length. Nonlineardevelopment of the instabilities with multiple eddies and shocklets, are simulated in a domainwith 2,048 finite volumes in x-direction and 1,024 finite volumes in the y-direction.

3 Shear instabilities in non-rotating flow

The first series of numerical simulations are conducted for the non-rotating shear flow withthe hyperbolic-tangent (TANH) base-flow velocity profile:

U (y) = U2 + 1

2(U1 − U2)[tanh

y

�sy + 1] (5)

As shown in Fig. 1, the free-stream velocity on one side of the base flow is U2 as y/�s → −∞,and on the other side is U1 as y/�s → +∞. The velocity difference U1 − U2 = � is thevelocity scale. The velocity gradient at the inflection Uy = 0.5�/�s defines the lengthscale �s . The hat symbolˆdenotes the value at the inflection where the velocity gradient is amaximum. The width δ = �/Uy = 2�s , to be referred as the maximum-gradient width, isdefined by the velocity gradient at the inflection. The depths are H1 and H2 on side 1 andside 2 of the free streams, respectively. The corresponding speeds of the gravity waves in thefree streams are

c1 = √gH1 and c2 = √

gH2. (6)

Fig. 1 Computation domain defined by the periodic boundary conditions at i = 1 and i = Imax and theradiation boundary conditions at y = y+ and y = y−

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The computational domain as shown in Fig. 1 is defined by the periodic boundary conditionsat i = 1 and i = Imax. The radiating boundary conditions,

v+ = c1h+ − H1

H1and v− = −c2

h− − H2

H2, (7)

are imposed so that the waves will escape without reflection at the boundaries y = y+ andy = y−. The dimensionless parameters for the development of the mixing layer are thevelocity ratio

� = U1 − U2

U1 + U2(8)

and the convective Froude number

Frc = U1 − U2

c1 + c2. (9)

The alternate to convective Froude number Frc and velocity ratio � is to use the free-streamFroude numbers Fr1 = U1/c1 and Fr2 = U2/c2 as the dimensionless parameters.

4 Linear and nonlinear development

There are two stages in the development of the instabilities. In the initial stage of the devel-opment, the amplitude of the instabilities is small and the nonlinear terms in the SWEs arenegligible. The linearized SWEs admit the normal-mode solution

[u′, v′, h′] = [u(y), v(y), h(y)] exp[ik(x − ct) − αt] (10)

in which k = 2π/λ = wave number, λ = wave length, c = wave speed and α = exponentialgrowth rate. This normal mode may be viewed as a traveling wave:

[u′, v′, h′] = [Au(y, t), Av(y, t), Ah(y, t)] exp[ik(x − ct)] (11)

The wave amplitude, [Au(y, t), Av(y, t), Ah(y, t)] = [u(y), v(y), h(y)] exp [-αt], of thetravel wave growth exponentially as follows:

α = 1

Au

dAu

dt= 1

Av

dAv

dt= 1

Ah

dAh

dt(12)

In the classical linear stability analysis (LSA), the exponential growth rate α is determinedas an eigenvalue problem of the linearized SWEs. In this paper, however, the exponentialgrowth rate in the linear stage is evaluated directly from the fully nonlinear simulation (FNS)using the SWEs. The FNS starts with a small lateral velocity, v′ of a specified wave lengthλ. The subsequent development of the v′-component of the velocity is used to calculatethe growth rate α. The maximum v′

max and the minimum v′min defines the amplitude Av =

|v′max−v′

min|. With |v′max−v′

min| as the amplitude of the instabilities,

α = 1

|v′max − v′

min|d|v′

max − v′min|

dt= d[ln |v′

max − v′min|]

dt. (13)

Figure 2 shows the linear and nonlinear stages of the development of the dimensionlessamplitude, |v′

max−v′min|/(U1 − U2), versus dimensionless time, tUy , in a semi-logarithmic

scale for a dimensionless wave number k∗ = 2 π �s/λ = πδ /λ = 0.393 in an unstableshear flow of convective Froude number Frc = 0.4. As shown in the figure, the initial growth

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Fig. 2 Linear and nonlinear stages of the development of the amplitude |νmax − νmin| with time t fora shear flow of convective Froude number Frc = 0.4 starting from a small perturbation of wave numberk∗ = πδ / λ = 0.393

for time Uyt < 190 is exponential when the nonlinear terms in the SWEs are negligible.The magnitude of the velocity fluctuations, u′ and v′, eventually become comparable withthe velocity of the base flow. For time Uyt < 190, the process becomes nonlinear. In thenonlinear stage of the instabilities, the base flow is modified by the disturbance of finiteamplitude.

Figure 3 shows the vorticity ζ ′ and depth h′ profiles over one wave length λ in the linearstage of the development when the amplitude of the instabilities is small. Figure 4 showsthe vorticity ζ ′ and depth h′ profiles over many waves length in the nonlinear stage of thedevelopment. At the low convective Froude number of Frc = 0.4, the mixing across theshear flow is due to the roll-up of the vorticity to form eddies, and coalescents betweenneighboring eddies. The mixing process at the higher convective Froude number is quitedifferent. At Frc = 0.8, the shear flow is characterized by local energy dissipation across theshock waves (hydraulic jumps) and the radiation of wave energy away from the shear flow.

5 Exponential growth rate

Figure 5 shows the exponential growth rate α obtained from the present fully nonlinearsimulations (FNS). The amplitude |v′

max−v′min| is calculated from the simulation data and α

is evaluated using Eq. (13). The dimensionless rate α/Uy as shown in the figure is a functionof the dimensionless wave number k∗ = π δ /λ. The convective Froude number Frc is theparameter. The present FNS are consistent with the LSA by Betchov and Szewczyk [3] andMichalke [26]. The maximum rate, αmax/Uy = 0.1898, was obtained at a dimensionlesswave number k∗ = 0.445 by Michalke’s [26] LSA for the limit case when Frc = 0. The datain Fig. 5 clearly show that maximum growth rate αmax/Uy to decrease with the increase ofthe convective Froude number Frc. The rate however is not dependent on the velocity ratio� (see definitions of Frc and � given by Eqs. (8) and (9), respectively).

The FNS was carried out for several velocity ratios: � = 0.1,� = 0.5 and � = ∞. Theopen symbols with the vertical bar in the figure denote the results obtained for � = 0.5.The open symbols with a horizontal bar in the figure are the results for � = 0.1. The othersymbols are for � = ∞. All three velocity ratios of � = 0.1, 0.5 and ∞ produced thesame results although the free-stream Froude numbers Fr1 = U1/c1 and Fr2 = U2/c2 arevery different. These calculations show clearly the sole dependence of the instability on the

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Fig. 3 Vorticity ζ ′ (top) and depth h′ (bottom) profiles in the initial linear stage of development in a non-rotating shear flow: a convective Froude number Frc = 0.4; b convective Froude number Frc = 0.8; and cTANH base-flow velocity profile

convective Froude number, not on the free-stream Froude number. The free-stream Froudenumbers nevertheless can be the relevant dimensionless parameters for shallow flows entirelydriven by friction [15,23].

6 Gas-dynamics analogy

The radiation of sound suppresses the shear instability. This fact has been known for sometime in gas dynamics [19,22,28,34,36]. Lin [22] studied the instability of shear flow betweenparallel streams in a gas using a convective reference frame moving with the instabilities. Heintroduced the convective Mach number for the instability as ratio of the velocity difference(U1 − U2) and the sum of the speeds of the sound c′

1 and c′2, in the free streams:

Mac = U1 − U2

c′1 + c′

2(14)

Papamoschou [28] used this convective Mach number to correlate experimental data. TheLSA results by Sandham and Reynolds [34] for ideal gas are also correlated with the con-vective Mach number. The lines in Fig. 5 are the LSA results of Sandham and Reynolds

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Fig. 4 Vorticity (top) and depth (bottom) profiles obtained from the simulations for eddies in non-rotatingshallow shear flow of convection Froude number Frc = 0.1 (left) and for shocklets in flow of Frc = 0.8(right). The color blue denotes the positive values of the vorticity and depth while and the red denotes thenegative values. Abrupt change in depth is observed across the shock waves (hydraulic jumps) in the flow withFrc = 0.8

Fig. 5 The dependence of normalized growth rate, α/Uy , on dimensionless wave number k∗ = πδ / λ andconvective Froude number Frc. The LSA results by Sandham and Reynolds [34] for ideal gas are included aslines in the figure for comparison

[34] correlated with the convective Mach numbers Mac = 0.01, 0.4, 0.8 and 1.2. These LSAresults for the gas are in close agreement with the present FNS for the instabilities in shallow

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waters at the convective Froude numbers of Frc = 0.01, 0.4, 0.8 and 1.2. The close agreementbetween the shear instabilities in shallow waters and those of the compressible gas howeverwas anticipated. The shallow-water equations are in fact identical to the 2D gas-dynamicequations when the specific heat ratio is 2 [21]. The connection between Froude and Machnumbers for the shallow flow dominated by friction was studied previously by Ghidaoui andKolyshkin [15] and Kolyshkin and Ghidaoui [23].

7 Instabilities’ energy and radiation

Insight into the effect of wave radiation can be studied further using the energy equation forthe instabilities. The linearized SWEs for the instabilities are:

∂h′

∂t+ U

∂h′

∂x+

(∂u′ H∂x

+ ∂v′ H∂y

)= 0 (15)

∂u′

∂t+ U

∂u′

∂x+ v′ ∂U

∂y− f v′ = −g

∂h′

∂x(16)

∂v′

∂t+ U

∂v′

∂x+ f u′ = −g

∂h′

∂y(17)

Multiplying Eq. (16) by u′ and Eq. (17) by v′, the sum of the two equations gives theinstabilities’ energy equation. Averaging the equation over the entire computation domain,gives

d K

dt= P + R (18)

where K = 1

2

(u′2 + v′2

)= kinetic energy of the instabilities, (19)

P = −u′v′ dU

dy= shear production, and (20)

R = −g

(

u′ ∂h′∂x

+ v′ ∂h′∂y

)

= radiation. (21)

The double bars above the variables denote the spatial average operation. The fractionalchange of the kinetic energy of the instabilities is the growth rate:

α = 1

K

d K

dt= P

K+ R

K(22)

This equation determines the same growth rate α as Eq. (13). The radiation term R is afunction of the gravity g and the depth gradients ∂h′/∂x and ∂h′/∂y. It is closely relatedto the radiation of the gravity waves. Figure 6 shows the magnitude of this radiation termrelative to the production term. The radiation is a negative contributes to shear instabilities.

The fraction R/P increases with Fr2c . The growth rate α/Uy therefore is reduced with Fr2

cdue to the increase share of the radiation over the production. The instabilities occupy onlya tiny fraction of the total energy in the shear flow. It grows exponentially in the initiallinear stage of its development. The energy flux radiated from the instabilities therefore isa function of time and space. It is related to and not equal to the radiation term given byEq. (21).

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Fig. 6 The increase in theradiation fraction relative toproduction, R/P , with theconvective Froude number Fr2

c

8 Earth’s rotation interference

To study the interference by the Earth’s rotation, our attention turns to shallow density cur-rents of large horizontal extend. In the Lower St. Lawrence River Estuary, horizontal tur-bulence as large as 30 km has been observed in an area of the estuary where the depthis 300 m [6,25]. The Gulf Stream in the Atlantic has a width of 60 km and a depth ofonly 600 m (Stommel 1951, 1965, [33]). The Jet Stream in the atmosphere is a densitycurrent hundreds of kilometers wide with a depth of no more than 10 km. The instabilityof the Gulf Stream in particular has received much attention [16,20,24,29]. In this paper,the Earth’s rotation effect on the instability of a shallow jet is considered as a one-layermodel of these currents. Figure 7a, b delineate the jet model of currents in the oceans and ofwinds in the atmosphere, respectively. The current U of the jet in the longitudinal directionproduces a Coriolis force, f U, in the lateral direction. For parallel flow to be possible, thedepth of the base flow, H , must vary in the lateral direction so that the pressure gradient,g(∂ H/∂y), is in geostrophic balance with the Coriolis force, f U . To study the current’sinstability, the base flow is perturbed by a small disturbance. The subsequent linear andnonlinear development of the instability is analyzed by fully nonlinear simulation using theSWEs.

9 Shallow jet in geostrophic balance

The base flow of the shallow jet as shown in Fig. 7c has a hyperbolic-secant (SECH) velocityprofile:

U (y) = � sech2 y

�S(23)

The velocity scale of the profile is �. The length scale is �S . The maximum of the velocitygradient at the inflection is Uy = 4

√3/9 �/�S . The width of the flow by the maximum

gradient is δ = �/Uy . The relation between the length scales for the SECH profile is�S = 4

√3/9 δ. The Rossby number is the ratio of Uy and f :

Ro = Uy

f(24a)

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Fig. 7 Shallow jet as a model for ocean currents and atmospheric wind in a rotating Earth: a ocean-currentmodel; b moisture-wind model; c the cyclonic and anti-cyclonic sides of the SECH-velocity profile; and d thedepth profile for geostrophic balance of the lateral pressure gradient with the Coriolis force

It also is the ratio of � and f δ:

Ro = �

f δ(24b)

For geostrophic balance, the Coriolis force must equal to the transverse gradient of thepressure force; i.e.,

− f U = gd H

dy, (25)

where g is the reduced gravity in the one-layer model. The depth profile required for thegeostrophic balance is

H(y) = H1 + H2

2− H1 − H2

2tanh

y

�S. (26)

Figure 7d shows the depth profile of the rotating current. Substituting the Eqs. (23) and (26)for U and H in Eq. (25) gives

Fr2c =

[3√

3(H1/H2 − 1)

8(√

H1/H2 + 1)2

]

Ro (27)

The convective Froude number Frc and Rossby number Ro therefore are related to each otheras a requirement for geostrophic balance. For the jet,

Frc = �

c1 + c2= �√

gH1 + √gH2

. (28)

The simulation presented in this paper is for the depth ratio of H1/H2 = 21. For this depthratio,

Fr2c = 0.417 Ro for H1/H2 = 21 (29)

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The small depth on side 2 of the jet is assigned for computational stability. It neverthelessapproximates the situationthat is found at the fronts between water masses of different den-sities in oceans, and at the fronts between air masses of different moistures and temperaturesin the atmosphere.

10 Vorticity and enstropy

Although Coriolis parameter is included in the SWEs, it does not explicitly appear in anyof the terms in the energy equation Eq. (18). The enstropy associated with the vorticityfluctuations of the instabilities however are explicitly affected by the Earth’s rotation. Webegin the derivation of the enstropy equation from the conservation of potential vorticity:

D

Dt

[f + ζ + ζ ′

H + h′

]= 0 (30)

where ζ = −Uy is the vorticity of the base flow and ζ ′ = v′x −u′

y is the vorticity fluctuationsof the instabilities. The potential vorticity is conserved when friction is ignored (see, e.g.,Chap. 4 of Cushman-Roisin [14]) for the derivation of the vorticity equation from SWEs).Subtracting the base flow and neglecting the nonlinear terms gives the Rayleigh’s equationfor vorticity fluctuations:

Dζ ′

Dt+ ε′

[f − Uy

H

]+ v′ H ∂

∂y

[f − Uy

H

]= 0. (31)

The dilation fluctuations and the base-flow potential vorticity are two important terms in thevorticity-fluctuation equation:

dilation fluctuation = ε′ = ∂ Hu′

∂x+ ∂ Hv′

∂y(32)

base-flow potential vorticity =[

f − Uy

H

]. (33)

The dilation fluctuation ε′ is important for its role associated with the waves. It is equal toDh’/Dt through the continuity equation. Without waves, the dilution fluctuation term and thesecond term of Eq. (31) would be zero. The rotational shear flow could not become unstableunless β = d f/dy is non zero according to Kuo’s [24] analysis of the β-effect. With waves onthe other hand, the shear instabilities would be suppressed by the radiation. The degree of thesuppression however depends on Coriolis parameter. Multiplying Eq. (31) by the vorticityfluctuation ζ ′ and then averaging over the entire domain of computation gives the following

equation for the production and radiation of the enstropy 12 ζ ′2 :

D

Dt

[1

2ζ ′2

]= Pζ + Rζ (34)

in which

Pζ = −v′ζ ′ H d

dy

[f − Uy

H

]= enstropy production (35)

Rζ = −ζ ′ε′[

f − Uy

H

]= enstropy radiation. (36)

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11 Linear instability of anti-cyclonic half jet

The jet has two sides as shown in Fig. 7c. The cyclonic side of the jet that rotates in the samedirection as the Earth is stable to perturbation. We therefore consider first the instability ofthe half jet on the anti-cyclonic side of the jet. The instability of the half jet is restricted tothe varicose mode, due to the solid boundary condition at the centerline of the jet. Figure 8ashows instability diagram for the exponential growth rate α of the anti-cyclonic half jetobtained from the numerical simulation. The diagram is a series of curves for the variationof α/Uy versus the dimensionless wave number k∗ = 2π�S/λ. Each curve is marked bya point on the curve to denote maximum growth rate. Figure 8b plots the maximum rateαmax/Uy for the Rossby numbers Ro = 0.5, 0.67, 1.0, 1.45, 2, 3, 4 and 6. The maximum rateαmax/Uy first increases and then decreases with Ro. The peak value [αmax/Uy]peak ∼= 0.054for the rotating half jet is remarkably close to rate of the non-rotating jet of constant depthobtained using the LSA by [4]: αmax/Uy ∼= 0.06 for the varicose mode of the nonrotating jetat the wave number k∗ = 2π�S/λ ∼= 0.5. Even more remarkably, this peak coincides with

the minimum of the wave-radiation energy fraction R/P and the minimum of the enstropy-

radiation fraction Rζ /Pζ as shown in Fig. 8c and d, respectively. The rotating half-jet ismost unstable in the window centered on a critical Rossby number of Roc = 1.45. Figure 9offers an explanation of the rotational interference at the critical Rossby number. The depthand velocity profiles are shown in (a) and (b). The potential vorticity ( f − Uy)/H in (c)changes sign on the anti-cyclonic side of the jet when the Rossby number Ro > 1 (i.e., when

Fig. 8 a Stability diagram for the anti-cyclonic half jet for Rossby numbers varying from Ro = 0.5 to 6. Thepoints on the curves mark the maximum growth rate αmax and the dashed line delineates the α-versus-k relationfor the critical Rossby number of Roc = 1.45. b The peaking of maximum rate in a window of instabilitycentered on the critical Rossby number of Roc = 1.45. c The suppression of the radiation fraction at thewindow due to rotation. d The depression of the enstropy radiation at the critical Rossby number Roc = 1.45over the window of instability

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Fig. 9 a Depth profile. b Velocity profile. c The (Uy − f )/H profile; it changes from a negative value (blue)

to a positive value (red) on the anti-cyclonic side of the jet when f < Uy . (d) The (Uy − f )/H profile when

f > Uy

Fig. 10 a Stability diagram for the geostrophic jet of the SECH velocity profile for Rossby numbers varyingfrom Ro = 0.33 to 25. The points on the curves mark the maximum growth rate αmax. The dashed linedelineates the α-and-k∗ relation for the critical Rossby number of Roc = 4. b The maximum rate αmax and itsdependence on the Rossby number Ro. c The depression of enstropy radiation at the critical Rossby numberRoc = 4. d The radiation energy fraction

f < Uy). The cancellation between the positive and negative parts of the potential vorticity( f − Uy)/H in the anti-cyclonic side of the jet on the average suppresses the wave radiationeffect. With the suppression, the energy available for instability is enhanced, thereby leadingto [αmax/Uy]peak ∼= 0.054 at the critical Rossby number Roc = 1.45. The waves are lesssuppressed outside the window of instability.

12 Linear instability of rotating full jet

Figure 10a shows the instability diagram for the full jet. Curves of the growth rate α/Uy

versus k∗ were presented for the Rossby numbers Ro = 0.33, 0.75, 1, 1.5, 2, 4, 8, 25 (cor-

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responding to convective Froude numbers Frc = 0.37, 0.55, 0.65, 0.79, 0.91, 1.29, 1.83 and3.23, respectively). The maximum rate αmax/Uy for each Rossby number Ro is marked bya point on the curve. Figure 10b shows the variation of the maximum rate, αmax/Uy , versusthe Rossby number, Ro. The peak of the maximum is [αmax/Uy]peak ∼= 0.10; it occurs at thecritical Rossby number of Roc ∼= 4. This peak value for the rotating full jet is lower thanthe value of αmax/Uy ∼= 0.21 for the sinuous mode, and higher than αmax/Uy ∼= 0.06 forthe varicose mode of the non-rotating jet (see, e.g., [10]). As shown Fig. 10c, the minimum

of the enstropy-radiation fraction, Rζ /Pζ , also occurs at the same critical Rossby number

Roc = 4. The energy-radiation fraction, R/P , of the rotating full jet however does not havea minimum (see Fig. 10d), as the stability is complicated by the cyclonic side of the jet.

13 Nonlinear transition across the window

The fully nonlinear simulations (FNS) determine not only the linear instabilities. It producesalso the nonlinear development of the instabilities when the base flow is modified by theinstabilities. The nonlinear development of the shallow rotating jet is quite fascinating asshown in Figs. 11, 12, 14, 15 and 16. In this series of the nonlinear simulation, the jetstarts with an initial Rossby number of Ro = 25 (with a corresponding initial convectiveFroude number of Frc = 3.23). A layer of red dye is placed initially along the jet centerlineas tracer for flow visualization. From left to right, Fig. 11a–d show the beginning of thenonlinear development at time Uyt = 150. The tracer red dye is seen in (a) perturbed by thepresence of the instabilities. The dilation pattern in (b) shows the waves radiating from theshear instability. The base-flow velocity profile in (c) and the depth profile in (d) are slightlymodified by the growth of the disturbance at this time. The maximum velocity of the meanflow drops slightly from the initial value of Um = � to Um = 0.8�. The width also increasesslightly at this point in time.

Figure 12 shows the brief and intense transition leading to the formation of eddies andshocklets. The mean flow of the shallow jet is now significantly modified due to the growthof the disturbance. At these times Uyt = 200 and 250, the maximum velocity drops toUm = 0.27� and 0.24�, respectively. The width of shear layer increases significantly. Fitting

(a) tracer c and u vector at time t = 150 (b) dilation ε at time t = 150 (c) U (d) h

Fig. 11 Nonlinear development of a rotating jet at time Uy t = 150 starting with an initial Rossby numberRo = 25 and depth ratio of H2/H1 = 21: a tracer concentration c and velocity vectors of the instabilities; bdilation ε′ pattern; c velocity of mean flow U profile; and d mean depth H profile

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(a) tracer c and u’ vector at time t = 200 (b) dilation ε at time t = 200 (c) U (d) h

(a) tracer c and u’ vector at time t = 250 (b) dilation ε at time t = 250 (c) U (d) h

Fig. 12 Brief and intense transition over a period of time from Uy t = 200 to 250: a tracer concentration cand velocity vectors of the instabilities; b dilation ε′ pattern; c velocity of mean flow U profile; and d meandepth H profile

the mean velocity by a SECH profile to estimate (Uy)modified, the width of the profile is thencalculated as δmodified = Um/(Uy)modified. The corresponding modified Rossby number is

Romodified = (Uy)modified

f= Um

f δmodified(37)

In the period of rapid nonlinear transition, the modified Rossby number drops sharply fromRomodified = 24 at Uyt = 100 to Romodified = 1 at Uyt = 250, as shown in Fig. 13a. Theintense and rapid transition is attributable to the approach toward and the cross over thepeak growth rate [αmax/Uy]peak at the critical Rossby number of Roc = 4. Figure 13b is areproduction of the maximum rate curves in Fig. 10b. It shows how the rate follows the pathof along the curves in the direction of the arrows as the modified Rossby number drops fromRomodified = 24 to Romodified = 1.

14 Post-transition period

The Rossby number maintains at a quasi-steady state value of Romodified ∼= 1, essentiallyunchanged over a long post-transition period from Uyt = 300–1,000 and beyond, as shownin Fig. 13a. Figure 14 shows the evolution of the instabilities in the post-transition periodfrom time Uyt = 300–500. The instabilities formed in the post transition is a complex mix

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Fig. 13 a Sharp drop of the modified Rossby number through the brief and intense nonlinear transition overa period of time from Uy t = 100 to 250; b The modified Rossby number along the path of instability in thedirection of the arrows during the rapid nonlinear transition

of eddies and shocklets. The sharp changes in velocity across the shock waves are shown in(a). The dilation ε′ pattern in (b) reveals the shock waves. The changes in the mean velocityprofile and mean depth profile in (c) and (d) are less dramatic and appear to approach aquasi-steady state at this period of time.

14.1 State of tranquility

The waves eventually diminish as the shear flow settles down to a state of tranquillity invirtual ageostrophic balance. Figure 15 shows the state of tranquillity at a much later time,Uyt = 3,000. The ring-like motions at time Uyt = 3,000 in Fig. 15 are markedly visiblebecause the tracer is entrapped in the rings in ageostrophic balance. The same rings can beidentified since their creation at time Uyt = 350. The pattern of the flow in the rings remainspractically unchanged over a long period of time, from time Uyt = 350–500, and then tomuch later time Uyt = 3,000. Similarly stable rings have been extensively studied in theGulf Stream of the Atlantic Ocean by Richardson [33] and others. Once formed, the GulfStream rings can remain in the ocean for years. The persistence of their motion is due tothe perfect balance between the pressure gradient and the centrifugal and Coriolis forces asanalyzed by Olson [27].

15 Summary and conclusion

Numerical simulations using fully nonlinear shallow-water equations have determined theshear instability in shallow flow in agreement with the classical linear stability analysis(LSA) by Michalke [26], Chu et al. [10], Sandham and Reynolds [34], and Jirka and co-author [11,12]. The radiation of gravity waves in shallow waters is a phenomenon analogousto the radiation of sound in gas dynamics. The convective Froude number in shallow watersis the analogous equivalent of the convective Mach number in gas dynamics. As in gasdynamics, radiation suppresses shear instability in non-rotating shallow flow. The Earth’srotation, however, alters the effectiveness of the radiation’s suppression. For rotating shearflow of sufficiently high convective Froude number, the wave radiation is restrained over anarrow window of instability at a critical Rossby number when the shear rate is comparable

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(a) tracer c and u’ vector at time t = 300 (b) dilation ε at time t = 300 (c) U (d) h

(a) tracer c and u’ vector at time t = 350 (b) dilation ε at time t = 350 (c) U (d) h

(a) tracer c and u’ vector at time t = 400 (b) dilation ε at time t U (d) h

(a) tracer c and u’ vector at time t = 500 (b) dilation ε at time t

= 400 (c)

= 500 (c) U (d) h

Fig. 14 a Velocity vectors and tracer pattern, b dilation pattern of the instabilities and c the mean velocityand d the depth profiles of the rotating jet in the post-transition period from Uy t = 300–500

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(a) tracer c and u’ vector at time t = 3000 (b) dilation ε at time t U (d) h= 3000 (c)

Fig. 15 Rings in ageostrophic balance as the flow approaches a state of tranquility at time Uy t = 3,000

to the rotation. Fully nonlinear simulation of rotating shear flow produces eddies and rings, asthe Rossby number of the mean flow drops sharply in a brief and intense exchange across thewindow of instability. The remnants of the exchanges are stable rings in perfect ageostrophicbalance. The persistent stability of the rings after the intense exchanges is remarkably similarto the persistence of Gulf Stream rings observed in the Atlantic Ocean (see e.g. [33]).

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