## Abstract

The axisymmetric intrusion of a fixed volume of fluid, which is released from rest and then propagates radially at the neutral buoyancy level in a deep linearly stratified ambient fluid, is investigated. Attention is focused on the development of self-similar propagation. The shallow-water equations representing the high-Reynolds-number motion are used. For the long-time motion, an analytical similarity solution indicates propagation with *t*^{1/3}, but the shape is peculiar: the intrusion propagates like a ring with a fixed ratio of inner to outer radii; the inner domain contains clear ambient fluid. To verify the similarity analytical prediction, a long-time finite-difference solution with realistic initial conditions was performed. To avoid accumulation of numerical errors, the problem was reformulated in terms of new variables. It is shown that the numerical solution has a ‘tail-ring’ shape. The ‘tail’ decays like *t*^{−2} and the ‘ring’ tends to the analytical similarity prediction. The initial geometry of the lock does not influence this result. Comparison with the non-stratified case is also presented. It was found that for the non-stratified case, there is a stage of propagation in which the intrusion has a similar ‘tail-ring’ form; however, this stage is only a transient to a self-similar shape which is different from that obtained for the stratified ambient.

## 1. Introduction

Intrusive gravity currents are formed when a given volume of fluid of constant density *ρ*_{c} and kinematic viscosity *ν* is released from a lock of height *h*_{0} into a vertically stratified ambient of density *ρ*_{a}(*z*) (figure 1). For simplicity, we assume that the intrusion is released at the level of neutral buoyancy. Gravity currents can play a role in industrial settings and will definitely play a major role in many large-scale natural situations as reviewed by Huppert & Simpson (1980) and Simpson (1997). We focus our attention on an axisymmetric geometry. The typical system configuration is sketched in figure 1. The propagation starts from the rest in a region of radial dimension *r*_{0} about the axis and the velocity has no lateral (azimuthal) component. We assume that the density of the ambient fluid varies linearly over the full depth of the container and that the Reynolds number *Re* of the flow is large.

The previous investigations (e.g. Hoult 1972; Grundy & Rottman 1985; Slim & Huppert 2004) were concerned mainly with the axisymmetric gravity currents released from behind a lock into a non-stratified homogeneous ambient. A closed one-layer shallow-water (SW) Boussinesq inviscid formulation was presented. In general, the solution of the resulting hyperbolic system is obtained by a finite-difference scheme. However, for the long-time developed motion, an analytical similarity solution has been derived. The self-similar result indicates radial expansion with *t*^{1/2}, where *t* is the time after release. However, it is not easy to compare this analytical prediction with the numerical results at large times after release, when the current becomes very thin and any numerical method becomes prone to round-off error. To overcome this loss of accuracy, it is convenient to reformulate the original SW problem in terms of the new dependent variables. This method was discussed by Grundy & Rottman (1985). The reformulated initial-value problem was solved by an explicit Lax–Wendroff method for *t*>1 (hereafter, the non-dimensional variables *t* and *r*, which will be defined in §§2 and 3, are used). Grundy & Rottman showed that for this case, the similarity solution is well approached at *t*≈e^{12} and that the finite-different results do not change for a very long time of approximately *t*≈e^{18}.

The recent investigations of axisymmetric intrusions released from behind a lock into a linearly stratified ambient were presented by Ungarish & Zemach (2007). They showed that for the large-time developed motion, an analytical similarity solution exists. The self-similarity result indicates radial propagation with *t*^{1/3}, but the shape is peculiar: the intruding fluid propagates like a ring with a fixed ratio of inner to outer radii. Inside the inner radius of the ring, there is a thin ‘tail’—residual layer of mixed fluid, whose thickness decreases like *t*^{−2}. The tail can also be characterized by the horizontal interface and linear radial velocity profile ∼*r.* To confirm this analytical approach, Ungarish & Zemach (2007) have solved the SW initial-value problem by a finite-difference scheme for 0≤*t*≤10. They showed that after an initial propagation to approximately 2.5 times the initial radius, at *t*≈5, the intrusion tends to a self-similar behaviour. However, the comparison was not sharp. Owing to the accumulation of errors, a long-time solution could not be obtained.

The ‘tail-ring’ form similarity solution of the axisymmetric stratified problem is unique and is different from the similarity solution of the non-stratified axisymmetric problem. As shown by Ungarish & Zemach (2002), very soon after the release, at *t*≈8, the axisymmetric non-stratified current develops a shape that is reminiscent of the current ‘tail-ring’ form of the stratified case. However, according to Grundy & Rottman (1985), the similarity solution for this case is different. If so, the ‘tail-ring’ stage of propagation of the non-stratified case is finite, and an additional stage between the tail-ring and the similarity solution stages should occur.

This behaviour of the non-stratified current indicates that the stage of stratified current should also be carefully investigated numerically for long periods. However, as for the non-stratified case, at large times after release, the current becomes very thin and the solution of the standard SW problem by numerical methods is expected to contain a large numerical error. To overcome this loss of accuracy, the problem should be reformulated.

In this paper, we take a closer look at the similarity solutions derived by Ungarish & Zemach (2007). We want to confirm numerically that these similarity solutions are the large-time limits of some class of initial-value problems associated with the SW equations. For this reason, the problem is reformulated in terms of long-time variables. In addition, we want to obtain some indication of how rapidly these similarity solutions are approached and how sensitive this rate of approach is to the initial and boundary conditions.

The structure of this paper is as follows. In §2, the SW equations of motion and the appropriate boundary conditions are presented. The long-time self-similar analytical solution of this system for a stratified ambient is discussed; the SW problem is reformulated in terms of new variables and is solved numerically. In §3, the propagation in the non-stratified ambient is discussed. The stratified and non-stratified cases are compared in §4. Finally, some concluding remarks are given in §5.

## 2. Formulation and SW approximations

We use the SW one-layer axisymmetric inviscid model. The ambient fluid is in the domain −*H*≤*z*≤*H* and is stably linearly stratified. The density increases linearly from *ρ*_{o} at *z*=*H* to *ρ*_{b} at *z*=−*H*. In the ambient fluid domain, we assume that *u*=*v*=*w*=0 and hence the fluid is in purely hydrostatic balance. The motion is assumed to take place in the intruding layer of fluid only, 0≤*r*≤*r*_{N}(*t*) and −*h*≤*z*≤*h*. The subscript N denotes the nose (front) of the intrusion. The density *ρ*_{c} of the intrusion is constant and is defined to be *ρ*_{c}=1/2(*ρ*_{b}+*ρ*_{0}). The initial flow-field configuration is symmetric with respect to the horizontal plane *z*=0. The following SW approximations are concerned with the inviscid and Boussinesq limits. In this case, the initial symmetry is expected to prevail also during the time-dependent propagation. It is therefore sufficient to consider the flow in the domain 0≤*z*.

We note that the buoyancy frequency of the unperturbed ambient is constant and is given by(2.1)where(2.2)and *g* is the gravitational acceleration.

The dimensional variables (denoted by asterisks) are scaled as follows:(2.3)where(2.4)where *r*_{0} and *h*_{0} are the initial length and half-thickness of the intrusion, respectively; *U*_{ref} is the typical inertial velocity of propagation on the nose; and *T*_{ref} is a typical time period for longitudinal propagation over a typical distance *r*_{0}. The typical Reynolds number is large and is defined by *Re*=*U*_{ref}*h*_{0}/*ν*, where *ν* is the kinematic viscosity.

We emphasize that hereafter the variables *r*, *z*, *u*, *t*, *h*, *H* and *p* are in dimensionless form, unless stated otherwise.

### (a) The governing equations

The equations of motion were formulated by Ungarish & Zemach (2007). The SW equations can be expressed in dimensionless form for the dense fluid variables: the height *h*(*r*, *t*) and the averaged longitudinal velocity *u*(*r*, *t*) by(2.5)

The appropriate boundary conditions are (i) the no-flow condition *u*(0, *t*)=0 at the centre, (ii) the kinematic condition at the nose (d/d*t*)*r*_{N}=*u*(*r*_{N},*t*), and (iii) the boundary condition for the velocity at the nose. The consensus is that the nose of a ‘real’ time- and space-dependent gravity current obeys a local, quasi-steady-state correlation between velocity and height. Ungarish (2005) showed that for an intrusion in a linearly stratified ambient, the proper condition is(2.6)where the Froude number *Fr* varies in a limited range about the value of unity and is actually a decreasing function of (*h*_{N}/*H*) only. A more rigorous justification of the nose condition is presented by Ungarish (2006) and Ungarish & Zemach (2007).

The experiments not only confirmed this qualitative behaviour, but also indicated that the theoretical value of *Fr*, derived by Benjamin (1968) to be for a deep ambient highly idealized motion, over-predicts (by typically 20%) the velocity in real circumstances. To reconcile theory with practice, in the present work we use for comparison the values of the Froude number in the range .

To strengthen the insights into the propagation of a deep intrusion, we consider the release of an initial volume from two different initial configurations: elliptical and cylindrical. The initial conditions are(2.7)

### (b) Similarity solution

Self-similar solutions play an important role in the SW analysis of gravity currents and intrusions in a non-stratified homogeneous ambient (Grundy & Rottman 1985; Gratton & Vigo 1994; Ungarish 2005).

According to Ungarish & Zemach (2007), for a deep intrusion and large values of *t*, the above-mentioned system of equations (2.5) with appropriate boundary conditions is satisfied by the following self-similarity solution(2.8)where(2.9)is a stretched radial coordinate that maps the radial domain [0,*r*_{N}(*t*)] into [0,1],(2.10)and *K* and *γ* are constants that represent the initial conditions. Obviously, *K* and (*t*+*γ*) must be positive for physically acceptable results.

The positive constant *K* is determined by volume continuity considerations to be(2.11)where *V*_{0} is the constant volume of the half-height (0≤*z*≤*h*) intrusion (per unit azimuthal angle), and is equal to 0.5 for the standard initial cylinder and to 1/3 for an initial ellipsoid. The constant *γ* can be calculated by matching the numerically computed *r*_{N} with the similarity form *K*(*t*+*γ*)^{1/3} at some time *t*_{match}. For example, for *Fr*=1.19 and *t*_{match}=5.5, we obtain *γ*≈−1.53 and −1.63 for the cylinder and the ellipsoid, respectively.

The solution (2.8) is valid only in the *y*_{1}≤*y*≤1 domain, where the intruding fluid propagates like a ring with a fixed ratio of inner to outer radii. We call this region the ‘ring region’. In ideal circumstances, the inner domain 0≤*y*≤*y*_{1} of the ring contains clear ambient fluid. In a more practical condition, inside the inner radius of the ring there is a thin tail—a residual layer of mixed fluid, whose thickness decreases like *t*^{−2}. The interface of the tail is horizontal and the corresponding velocity increases linearly with *r* (as first approximations). The analytical solution of (2.5) in the tail is given by(2.12)where *C*_{1} and *C*_{2} are positive constants of order unity. The fit at *t*≈8.0 to the numerical results for *Fr*=1.19 yields *C*_{1}=1.194 and *C*_{2}=0.194 for the cylinder. The fit at *t*≈14.0 for the ellipsoid gives *C*_{1}=1.946 and *C*_{2}=0.815 for *Fr*=1.19. We note that (2.12) is an exact solution of (2.5); formally, this can also be referred to as a self-similar inner-region behaviour (for large times).

To patch these two regions, we derive the analytical expression of their meeting point *y*_{M} and its dependency on time. By assuming that the height is continuous at *y*_{M}, the value of *y*_{M} can be obtained. This assumption is supported by the behaviour of the numerical solution performed by Ungarish & Zemach (2007). Thus, from (2.8) and (2.12), using *h*(*y*_{M−}, *t*)=*h*(*y*_{M+}, *t*), we obtain(2.13)and ∀*t*≥*C*_{2}/2 the function is a decreasing function, which can be limited by . A short calculation shows that (i) for *Fr*>1, the long-time solution conserves its ‘tail-ring’ form (the case *Fr*=1 is different from the rest, since for this case *y*_{1}=0 and the tail region vanishes), (ii) the value of *y*_{M} is always greater than *y*_{1,} and (iii) lim_{t→∞}*y*_{M}=*y*_{1}.

To confirm this analytical approach, Ungarish & Zemach (2007) have solved the SW initial-value problem by a finite-difference scheme for 0≤*t*≤10. They showed that after an initial propagation to approximately 2.5 times the initial radius, at *t*≈5, the intrusion tends to a self-similar behaviour.

However, Ungarish & Zemach's work could not consider the solution for a very long time due to loss of accuracy in the numerical solution of (2.5). To determine the large-time solution of this problem and to improve the comparison with the analytical similarity results, we found it necessary to reformulate the problem in terms of new variables.

### (c) Numerical solutions

We follow the method used by Grundy & Rottman (1985) for the non-stratified problem. Here, we develop an appropriate solution for the intrusion in a stratified ambient.

The original system of equations (2.5) with appropriate boundary and initial conditions (2.7) is reformulated here in terms of the new independent variables(2.14)and the new dependent variables *R*, *H* and *U*, defined by(2.15)where *K* is a constant given by (2.11).

The original equations were subjected to the following transformation:(2.16)where . Here, *R*′=d*R*/d*T* and it is expected to approach zero for *T*→∞.

The governing equations (2.5) now become(2.17)

The boundary conditions are(2.18)

The appropriate initial conditions (2.7) at *T*=0 (*T*=0 corresponds to *t*=1) are(2.19)

This system of hyperbolic equations can be written in the characteristic form(2.20)along the characteristic curves specified by(2.21)The SW system (2.17) with the appropriate boundary (2.18) and initial (2.19) conditions is solved numerically. The boundary conditions for *H* at *y*=1 and 0 are calculated for each new time step from the balances on the characteristics *C*_{±}. Here, we employed a finite-difference McCormack scheme (e.g. LeVeque 1992).

We chose to solve the problem for Froude number *Fr*=1.19. This value of *Fr* corresponds to a semi-empirical correlation suggested by Huppert & Simpson (1980) for deep currents. Various previous results were presented for this value of Froude number and were in a good agreement with the theory or experiments (see Ungarish & Huppert 2004; Ungarish 2005 and others). In particular, Ungarish & Zemach (2007) showed that there is a good agreement between the numerical solution of the Navier–Stokes equations and SW prediction of the axisymmetric intrusion for this value of Froude number.

The choice of variables keeps the new variables (2.14) and (2.15) within reasonable ranges as follows. Obviously, 0≤*y*≤1 and 0≤*T*, and we note that moderate values of *T* can be used (since *T*=2 corresponds to *t*=e^{6}≈403). The dependent variables were found to be in the range 0≤*H*<2, 0≤*U*≤2.5 and 0.6≤*R*≤1.1. This, as expected, allows us to calculate accurate results, which can be compared directly with the similarity forms (2.8).

### (d) Results and comparisons

The numerical results for the (transformed) SW equations were obtained using 4000 grid points and a time step of 5×10^{−6}. The comparison between the SW and similarity solutions for *Fr*=1.19 and cylindrical initial configuration is shown in figures 2 and 3.

Figure 2 shows the plots of *H*(*y*, *T*) and *U*(*y*, *T*) at several values of *T*. The similarity solution is shown as a bold solid curve. Figure 3 shows the plot of the front position *R*(*T*) as a function of *T* and the similarity solution is shown by the dashed line.

Figure 2 shows that at *T*≈1.0, the similarity ‘tail-ring’ form is already obtained. By the time *T*≈2.0 (*t*≈e^{6}), the numerical solution is very close to the analytical similarity solution. For *T*>2.0, this numerical solution changes very slowly and actually coincides with the numerical solution at *T*≈2.0, except, to a very small region near *y*=*y*_{1}. It can be noted that at *T*=2.24, the numerical solution passes the similarity solution; however, running the simulation to longer times probably would show that the solution backs and oscillates about the similarity solution.

Figure 3 shows that the solution *R*(*T*) is close to the similarity result 1 for *T*>1.1 (approximately) and that the agreement improves for *T*>1.5.

The comparison between the analytical and numerical results for the position of the meeting point *y*_{M} is shown in figure 4 for *Fr*=1.19. The numerical values of *y*_{M} were calculated according to the location of the velocity jump (figure 2). The function *y*_{M} decreases with *T* and its numerical value approaches *y*≈0.52 when *T*→∞. The comparison shows that the numerical results confirm the analytical estimate (2.13) and deviate from it by approximately −3%.

To strengthen the insights into the propagation of a deep intrusion, we also considered the release of an initial ellipsoid volume of mixed fluid, i.e. for 0≤*r*≤1 at *T*=0 (*t*=1). Figures 5 and 6 show that the essential propagation is similar to that of the previous cylindrical problem. The initial spread is slightly delayed by the fact that the height of the nose must develop from zero, but the tendency to the similarity shape at *T*>1.6 is evident.

## 3. Non-stratified case

For the non-stratified case (*ρ*_{a}=const.), the motion of the current is described by the SW equations (Grundy & Rottman 1985)(3.1)For this configuration, the dimensional variables (denoted by asterisks) are scaled as follows:(3.2)where(3.3)*r*_{0} and *h*_{0} are again the initial length and half-thickness of the intrusion, respectively; *g*′=|(*ρ*_{c}/*ρ*_{a})−1|*g*, *U*_{ref} is the typical inertial velocity of propagation on the nose; and *T*_{ref} is a typical time period for longitudinal propagation over a typical distance *r*_{0}. The typical Reynolds number is *Re*=*U*_{ref}*h*_{0}/*ν*, where *ν* is the kinematic viscosity.

Again, we consider the upper half *z*≥0. This corresponds to a gravity current on the *z*=0 boundary. To create an intrusion, we need *ρ*_{a}>*ρ*_{c} in the lower half, with the same |*ρ*_{c}−*ρ*_{a}| like in the upper half. In the lower half, we shall have a mirror image. In this case, the intrusion is in the two-layer ambient and not in a continuously stratified ambient.

We emphasize that hereafter the variables *r*, *z*, *u*, *t*, *h*, *H* and *p* are in dimensionless form, unless stated otherwise. The appropriate boundary conditions are as before: the no-flow condition *u*(0, *t*)=0 at the centre and the dependency of at the nose.

The initial conditions are identical to (2.7) and are essentially similar with the theoretical results obtained for idealized infinite systems by Benjamin (1968).

We note that the dimensionless equations of motion (2.5) and (3.1) are the same for both stratified and non-stratified cases, except the different second term in the momentum equation. The additional difference between the problems is in the boundary condition for *u*_{N}.

The self-similarity solution of the non-stratified problem, suggested by Hoult (1972), can be expressed as(3.4)where again(3.5)and(3.6)where *V*_{0} is the constant volume of the intrusion (per azimuthal angle) and is equal to 0.5 and 0.3 for the standard initial cylinder and an initial ellipsoid, respectively. We emphasize that (3.4) is valid for the full domain 0≤*y*≤1. We note that for , see (3.4), the interface profile attains negative values near the centre. This type of behaviour for large *Fr* has been pointed out by Gratton & Vigo (1994). Formally, this limits the applicability of the solution to some *y*≥*y*_{2}≥0 when . However, since for the Boussinesq currents considered here such values of *Fr* are physically impossible, this limitation can be ignored.

The original problem is reformulated here in terms of new variables. Following Grundy & Rottman (1985), in this section we use dependent variables *H*(*y*, *τ*) and *U*(*y*, *τ*), which are defined as(3.7)and the independent variables(3.8)

Consequently, the original equations (3.1) now become(3.9)This system can be written in the characteristic form(3.10)along the characteristic curves specified by(3.11)

The boundary conditions become(3.12)The appropriate initial conditions are(3.13)

Following Grundy & Rottman (1985), we used a Lax–Wendroff two-step scheme with 600 grid points and a time step of 3×10^{−3} to obtain the numerical results for the SW non-stratified equations. Results of the numerical calculations for *Fr*=1.19 case are shown in figure 7. The computed height *H*(*y*, *τ*) and velocity *U*(*y*, *τ*) profiles are showed as the functions of *y* for various values of *τ*=0.5, 1.0, 2.1, 6.0 and 9.0. Two main stages of propagation can be distinguished for the non-stratified case: first, the ‘tail-ring’ stage, which is similar to the stratified case, takes place between *τ*≈1 and *τ*≈2.2. However, here the tail shrinks: the matching point, , between the tail and the ring propagates backward and finally at *τ*≈2.2 it arrives at *y*=0 and the tail disappears completely. Approximately at *τ*=6, the solution approaches the second main stage—1 h similarity non-stratified solution (3.4), which has a rather different form without the ‘tail-ring’ behaviour. Figure 8 shows that the solution *R*(*τ*) is close to the similarity result 1 and *R*′(*τ*) oscillates about the similarity result 0.

To further confirm the previous results, we also consider the non-stratified case with *Fr*=1.0. The numerical results for this case are presented in figure 9. It is shown that the ‘tail-ring’ stage takes place for 1≤*τ*≤1.6 and the similarity solution (3.7) is closely approached at *τ*≈6.

This case is also discussed by Grundy & Rottman (1985) and can be compared with our results. Grundy & Rottman rescaled the original SW problem by the independent variables which were, however, different from our rescaling (3.8) and were defined by(3.14)The definition of the variable *y* defined by (3.8) keeps the variable *y* in the 0≤*y*≤1 domain, while for the variable *ξ*, defined by (3.14), the values *ξ*>1 are permitted. Grundy & Rottman solved the rescaled system of equations numerically for the dependent variables *H* and .

The graphs of *H*(*ξ*) and *R*(*θ*) presented by Grundy & Rottman (figures 3*b* and 4*b* in that paper) are in excellent agreement with our results. However, no graphs of *U* were shown by Grundy & Rottman and the ‘tail-ring’ stage of propagation was not discussed. These are, to our best knowledge, new results.

Grundy & Rottman noted that the solution *R*(*θ*) oscillates about the similarity result 1. The same behaviour was observed in our results for the *Fr*=1.0 and 1.19 cases. From figure 8*a*, it can be calculated that for *Fr*=1.19, as expected, the amplitude of the oscillations decreases with time as *τ*^{−0.5}. In this case, the behaviour of *R*(*τ*) can be approximated by *R*(*τ*)=0.02*τ*^{−0.5}sin(*ωτ*)+1, where *ω*≈2.24. The velocity of the decay is shown in figure 8*b*.

To strengthen the insights into the propagation of a deep non-stratified intrusion, we considered the release of an initial ellipsoid volume of mixed fluid, i.e. for 0≤*r*≤1 at *t*=0. (This has not been considered by Grundy & Rottman.). Figure 10 shows that the essential propagation is similar to the previous cylindrical problem. The initial spread is slightly delayed by the fact that the height of the nose must develop from zero, but the tendency to the similarity shape at *τ*>2.2 is evident. As before, the solution oscillates about the similarity result *R*(*τ*).

Consider the ‘tail-ring’ stage of the non-stratified case propagation. This stage starts at some time *t*_{TR} and finishes at some finite time *t*_{TRend}, as estimated below. During this stage, the solution (2.12) is a good approximation for some range (with the values of *C*_{1} and *C*_{2} as different from those calculated for the stratified case). Actually, note that this is an exact solution of (3.1). On the other hand, the similarity solution (3.4) occurs for . Again, by patching these solutions, we have the following analytical expression for :(3.15)and ∀*t*≥*C*_{4} the function is the decreasing function, which can be limited by . The ‘tail-ring’ stage must end at , when approaches zero and the tail part disappears.

The fit to the numerical Lax–Wendroff results at *t*≈10 for *Fr*=1.19 yields values of *C*_{3}=0.5 and *C*_{4}=0.3. The numerical value of , which was obtained from the location of the velocity jump, is in fair agreement with the analytical result.

## 4. Discussion

The analytically predicted similarity form of the axisymmetric stratified problem displays a peculiar ‘tail-ring’ structure for . The analytical similarity solutions were confirmed by the numerical results of the SW equations for long times. In both the stratified and non-stratified cases, a ‘tail-ring’ shape appears at approximately *t*=2 and in both cases, the thickness of the tail decreases like *t*^{−2}. However, for the stratified problem, the length of the tail increases (the patching point between the tail and the ring, *y*_{M}, is near ≈*y*_{1} and hence *r*_{M}=*y*_{M}*r*_{N}(*t*) increases). On the other hand, for the non-stratified case, the length of the tail decreases until the patching point approaches zero and the tail disappears. Thus, for the non-stratified case, the ‘tail-ring’ form appears only for a finite period of time during the transition to the final similarity solution. To compare the behaviour of the tail thickness for the stratified and non-stratified cases, the graph of *ht*^{2} versus *t* is shown in figure 11 for *Fr*=1.19. For this comparison, the numerical value of *h* was taken as an average value of the tail thickness in the 0≤*y*≤*y*_{M} (or for the non-stratified case) domain. For the stratified case, the function *ht*^{2} tends to 1.2, which confirms the analytical expression of the tail, given by (2.12). However in the non-stratified case, this function is also constant, ≈0.5, until the patching point approaches zero at t≈70 and then the tail vanishes and the ‘tail-ring’ form of the current disappears.

Finally, it can be noted that the effect of the internal gravity waves, which is ignored by the SW approximation, is very small in the axisymmetric geometry (Ungarish & Zemach 2007), in contrast to the two-dimensional rectangular counterpart problem (Flynn & Sutherland 2004; Ungarish 2005).

## 5. Conclusions

The propagation of an axisymmetric intrusion of a given volume released at the neutral buoyancy level in a stratified ambient was considered. We used a new analysis, based on a one-layer SW closed formulation. The previous SW results presented by Ungarish & Zemach (2007) show that after an initial propagation to approximately 2.5 times the initial radius, the intrusion approaches a self-similar behaviour with a unique ‘tail-ring’ form. On the other hand, analysis of the non-stratified intrusion shows that for this case there is also a stage of propagation when the intrusion has a ‘tail-ring’ form, but the analytical similarity solution for this case is quite different and a tail in the very long-time behaviour is unacceptable. Our investigation elucidated the details of the approach to similarity in these cases.

To verify the similarity analytical prediction long time after release, the SW problem was reformulated in terms of new variables. The advantage of this formulation is that the variables are found in the reasonable ranges and can be compared directly with the similarity solutions. Predictions were obtained for realistic cylindrical and elliptic lock geometries; realistic initial and boundary conditions. Various values of the Froude number were used in the domain .

The ‘tail-ring’ form of the current, obtained from the numerical solution of the transformed SW problem, is in good agreement with the analytical expression of the similarity solution (for both the cylindrical and elliptical initial geometry of the current).

The comparison of the results obtained for the stratified intrusion with the classical non-stratified case shows that in the latter case the ‘tail-ring’ shape appears only as a transient stage of propagation. The similarity behaviour of the current in this case is quite different from the former one without the ‘tail-ring’ behaviour. The solution of the non-stratified case is an extension of the work of Grundy & Rottman (1985). Some new results were presented here, in particular: (i) the similarity solution was verified by the numerical run of elliptical initial geometry for that case, (ii) the graphs of velocity were presented, and (iii) the *Fr*=1.19 was discussed.

After a spread to a relatively large radius, the intrusion is expected to become very thin and slow. At this stage, in a real fluid, the effects of viscosity, mixing and wave influence are expected to become dominant. This requires a separate investigation.

To our best knowledge, there are presently no experimental verification of the flow discussed in our paper. The clarification of this issue is left for future work.

## Footnotes

- Received November 7, 2006.
- Accepted May 14, 2007.

- © 2007 The Royal Society