## 1. Introduction

The surface quasigeostrophic (SQG) model (Held et al. 1995) is an approximation of the motion of a rapidly rotating stratified fluid near a horizontal boundary. It represents planar advection of the boundary temperature field under the assumption of uniform interior potential vorticity. This is in essence a two-dimensional system whereby the streamfunction is determined entirely by the boundary temperature field via a Green’s function proportional to 1/*r*. Applications of the model include the evolution of tropopause undulations (Juckes 1994; Juckes 1995; Muraki and Snyder 2007) and near-surface oceanic vortices (Lapeyre and Klein 2006). This type of dynamics has also been studied by Blumen (1978) under the name of uniform potential vorticity flow.

Carton (2009, hereafter C09) provides a numerical study of smooth shielded vortices in the SQG model and compares the evolutions with those of unstable vortices under the more familiar two-dimensional Euler dynamics. Here the complementary analytic study is presented in which linear growth rates of perturbations are calculated for a simple class of shielded SQG vortex, the shielded surface Rankine vortex. By this we mean the SQG analog of the usual shielded Rankine vortex of two-dimensional Euler dynamics: an inner circle of temperature *θ* = *θ*_{0} surrounded by an outer annulus of temperature *θ* = *θ*_{1}. In the following we use the same notation and conventions as C09, which should also be referred to for a full description of the model and its applications.

The shielded surface Rankine vortex provides the simplest analytically tractable example of an unstable SQG vortex. Furthermore, the corresponding two-dimensional Euler problem is well studied (Flierl 1988) so that this setup provides a fruitful source of comparisons between the two systems. The analytical treatment has become possible since Harvey and Ambaum (2010b) derived the dispersion relation for a single-step surface Rankine vortex, that is, a circular patch of uniform temperature anomaly analogous to the Rankine vortex of two-dimensional Euler dynamics. We review the methodology and the main results we require from this study in section 2 and then analyze the shielded problem in section 3. A comparison is made with the results of C09 in section 4 and conclusions are provided in section 5.

## 2. Single-step surface Rankine vortex dispersion relation

*a*is the patch radius and

*θ*

_{0}is the temperature anomaly of the vortex, is derived in Harvey and Ambaum (2010b). Here we review the methodology and the main results from that study. It is shown there that the azimuthal velocity field corresponding to (1) can be written as

*J*is the Bessel functions of the first kind. Figure 1a illustrates (1) and (2) graphically. Note the logarithmic singularity in

_{n}*u*

*r*=

*a*, which we discuss below.

*φ*is the azimuthal coordinate, and then linearizing the equation for material advection of the boundary for small

*η*. The linearized equation of motion is

*υ*′(

*r*,

*φ*,

*t*), the linearized perturbation radial velocity field, is derived in Harvey and Ambaum (2010b) as

*r*=

*a*, but the combination in (5) is regular. The velocity singularity is a generic feature of temperature discontinuities under SQG inversion. However, the singular component of the velocity is always aligned along the discontinuity and the normal component remains finite. Therefore, the evolution of the temperature field remains relatively slow (Held et al. 1995). This alignment is apparent here through the cancellation of the singularities in (5), as discussed in detail by Juckes (1995). The justification for using this basic state, despite the qualitative difference with the smooth profiles of C09, comes from the fact that wave propagation on a slightly smoothed discontinuity is regular in the sharp edge limit, as shown by Juckes (1995).

*u*

*υ*′ from (2) and (6) in (5) gives the dispersion relation we seek:

*r*in that case, introduces a factor

*k*

^{−1}into the integrand of (3) and the integrals can then be evaluated as

*a*/2

*n*(Gradshteyn and Ryzhik 2000) so (7) reduces to the standard result:

*q*

_{0}is the vorticity value of the patch. An important difference between (9) and (8) is the

*a*in the prefactor: the phase speeds of perturbations

*c*(=

_{p}*ωa*/

*n*) on the SQG vortex are independent of the vortex radius. This is to be expected on dimensional grounds since

*θ*has the dimension of a velocity field whereas the vorticity

*q*in the two-dimensional Euler system has the dimension of a frequency.

## 3. Shielded surface Rankine vortex dispersion relation

*a*<

*b*are the radii of the temperature jumps and

*θ*

_{0}and

*θ*

_{1}are the inner and outer temperature anomalies, respectively. Through linearity and (2) the corresponding basic-state velocity field is

*η*and

*ν*. The result is a pair of equations analogous to (5):

*u*

*r*) is given by (11) and

*υ*′(

*r*,

*φ*,

*t*) is likewise a linear sum of contributions from each boundary of the form in (6):

*u*

*υ*′, from (11) and (15) into (13) and (14), and write the system in matrix form for convenience. First we define the nondimensional parameters:

*E*(1/

_{n}*λ*) =

*λE*(

_{n}*λ*). The system in (13)–(14) can then be written as

*α*is defined in (8). The matrix

_{n}*θ*

_{0}/

*a*as the dimensional frequency scale for consistency with the study of Flierl (1988). That is, we imagine a central vortex of amplitude

*θ*

_{0}and radius

*a*and ask how the different “shields” modify the vortex behavior. An alternative, which we discuss further below, is to focus on the filament-like nature of the instability. That is, consider the outer annulus as a circular filament of width

*b*−

*a*, the instability of which is modified by the circular geometry and the presence of the central vortex.

*λ*,

*μ*) space is shown in Fig. 2a for several wavenumbers. To interpret the figure we note that the SQG analog of the Rayleigh theorem requires a radial temperature profile to contain regions of opposing gradient for growing normal modes to exist (see C09). These are only present in (10) if

*μ*> 1 or

*μ*< 0 and the regions of normal mode growth in Fig. 2a are indeed contained within these regions.

*q*

_{0}and an outer annulus with vorticity

*q*

_{1}. The two plots have many qualitative similarities, the main difference being that the

*n*= 2 mode is stable for all

*q*=

*q*

_{1}/

*q*

_{0}> 0 in the two-dimensional Euler case, whereas for the SQG system there is a region of parameter space with

*μ*> 0 where the

*n*= 2 mode is unstable. Another difference is that for the SQG case the boundaries of stability do not all continue to large

*λ*. For the two-dimensional Euler case the boundaries of stability all tend toward

*q*= (

*n*− 1)/(

*n*− 2) at large

*λ*(see Flierl 1988) whereas for the SQG system modes

*n*= 2–4 satisfy

*n*≥ 5 (for which

*α*> ½) are stable for all

_{n}*μ*at

*λ*larger than a critical value given by the solution to

*λ*

_{c}^{2}

*E*

_{1}(

*λ*) =

_{c}*α*. This expression is found by noting that the stability boundary is given by the implicit equation

_{n}*λ*≫ 0 we find the expression in (20), where we have also used the asymptotic result derived in Harvey and Ambaum (2010b), that is,

*E*~

_{n}*C*/

_{n}*λ*

^{n}^{+1}with

*C*

_{1}= ½.

The normal modes of (19) are special perturbation configurations that preserve their shape in time. In general, unstable solutions to equations of the form in (18) undergo an initial period of transient development during which they align toward the growing normal mode shape [see Farrell and Ioannou (1996) and Carton et al. (2010) for a full description of such nonmodal behavior]. The main result is that the transient development can exhibit a temporary period of growth at rates larger (or smaller) than the normal mode growth rate, depending on the initial configuration of the perturbation [i.e., *η*(0) and *ν*(0)]. Similarly, stable solutions can exhibit periodic cycles of transient growth and decay that, while not causing a long-term net growth to perturbations, may temporarily increase the perturbation size.

*N*(

_{s}*t*)/

*N*(0) is the singular mode amplitude scaled by the initial perturbation amplitude. It can be shown that (Farrell and Ioannou 1996)

*N*has the following asymptotic limit for small times

_{s}*F*

_{12}and

*F*

_{21}are the off-diagonal elements of

*N*(

_{s}*t*) ~ 0, otherwise. The figure shows these two limits graphically as well two intermediate times. It is consistent with the notion that nonmodal disturbances of wavenumbers within the range of growing normal modes undergo a transient period of growth potentially larger than the corresponding normal mode growth rate during which they adjust toward the normal mode shape, followed by what is effectively normal mode growth. Nonmodal disturbances outside the range of the growing normal modes continually undulate around their initial configuration resulting in a

*σ*, which decays in time. The ridges, particularly apparent in Fig. 3c, are a result of this undulatory nonmodal behavior.

_{s}## 4. Completely shielded vortices

Figure 4a shows the normal mode growth rates of perturbations for completely shielded vortices as a function of *λ*. The SQG vortices display the familiar behavior also observed in the two-dimensional Euler case, through theory and experiment, that the wavenumber of the fastest-growing normal mode is dependent on the vortex profile, with progressively higher modes being relevant for smaller *λ*, or larger *μ*. The increase in growth rate as *λ* → 1 is consistent with a consideration of filament instability, as we show below. First we compare the growth rates to those of the C09 study.

*r*is the radial coordinate,

*R*and

*α*are positive constants, and Θ

_{0}(

*α*) is a normalization factor such that the corresponding azimuthal velocity field

*u*

*r*) satisfies max(

*u*

*u*

_{0}. These profiles consist of a central temperature maximum surrounded by an annulus of negative temperature anomaly, as illustrated in Fig. 5. We approximate them as shielded surface Rankine vortices by choosing suitable values for

*θ*

_{0},

*θ*

_{1}and the sizes of the inner and outer regions.

*θ*

_{0}and

*θ*

_{1}are chosen so that both the total integrated temperature anomaly is zero and the integrated temperature anomaly in the range

*r*<

*a*equals that of the smooth profiles. The choice is of course not unique, but it suffices to provide a qualitative comparison between the systems. Figure 5 illustrates this choice of parameters for several values of

*α*. The differences between the two growth rates is expected to be sensitive in particular for small

*α*where the smooth profiles differ most from the Rankine profiles.

_{0}and

*α*are as in (25). Figure 4b shows the corresponding normal mode growth rates, Im(Ω

*), as a function of*

_{n}*α*. Note that the data plotted here are identical to those of Fig. 4a except for the choice of scalings.

Also shown in Fig. 4b are the numerically obtained growth rates from C09 for the corresponding smooth cases. There is a clear similarity between the two cases. The main difference is a shift toward higher *α* values in our figure since the first unstable mode appears near *α* = 3, whereas there are unstable modes from *α* = 2 in C09. The magnitude of the growth rates, however, is comparable between the two cases.

*λ*close to 1, or equivalently large

*α*, is consistent with the nature of filament instability in SQG dynamics. For comparison, a straight temperature filament of width

*L*with a “top hat” profile, analogous to the Rayleigh problem of two-dimensional Euler dynamics, is unstable with normal mode growth rates given by

*F*is a function of the nondimensional wavenumber

*κ*=

*kL*(Juckes 1995). As such, we define a scaling factor:

*F*)

*S*≈ 0.1292

*S*. We expect a similar result to hold for the two-dimensional Euler system, but in that case there is no length scale so the instability growth rates will just be proportional to the geometric mean of the two vorticity jumps.

*L*(i.e.,

*λ*→ 1 with

*b*−

*a*=

*L*). The wavenumber

*n*is scaled to coincide with the wavenumber

*κ*of the filament in this limit:

*S*of (33). The growth rates do indeed appear to coincide with those of the straight filament in the limit of

*λ*→ 1. The correction for larger

*λ*shifts the growth rate curve to slightly larger

*κ*

_{equiv}values on average but, consistent with the discussion above, does not alter the maximum growth rate by very much. This shift in the maximum growth rate seems to be purely a geometric effect of the curvature.

## 5. Discussion and conclusions

The linear stability of the SQG shielded Rankine vortex has been investigated analytically. The results show that the behavior found numerically by C09 for a class of similar, but smooth, temperature profiles is captured well by this simpler model. In addition, the study provides a comparison to the well-studied two-dimensional Euler case.

The instability is shown to be related to filament instability in the relevant limit and is remarkably similar even far from this limiting case. The interesting differences to the two-dimensional Euler case are a result of the scaling between the advected quantity and the flow field, which is apparent in the growth rate of the filament instability. Harvey and Ambaum (2010a) discuss this further in the context of a filament under strain and shear.

As a check of the theory, nonlinear simulations of the completely shielded vortices have been run at high resolution with a pseudospectral code. This was achieved by (very) slightly smoothing the vortex profile discontinuities. The experiments showed vortex instability on the linearly most unstable mode within a shift toward larger *λ* of order 0.05. In a similar fashion to the corresponding two-dimensional Euler case of Morel and Carton (1994), the numerical experiments did not evidence nonlinear stabilization into multipoles. Instead the outer annulus breaks nonlinearly into smaller vortices that then pull the central vortex apart and disperse as dipoles. However, as in the two-dimensional Euler case, stable multipoles can be achieved if an annulus of zero temperature is introduced between the core vortex and the active periphery.

An interesting problem following on from this study is to investigate the corresponding baroclinic problem, that is, to consider two vertically separated SQG temperature patches of opposite sign. This can be done either in an unbounded fluid, as an approximation to an oceanic heton, or confined by horizontal boundaries, as a model of atmospheric baroclinic instability on the hemispheric scale. This would put the model of Eady (1949) into a more realistic setting. A similar framework could be applied in either case.

## Acknowledgments

XC acknowledges financial support for this study from DGA under program REI COMINO and from UBO.

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