Time-dependent topographic meandering of a baroclinic current

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Dynamics of Atmospheres and Oceans, 1 (1976) 127- 161 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

TIME-DEPENDENT TOPOGRAPHIC MEANDERING OF A BAROCLINIC CURRENT

SULOCHANA GADGIL

Indian Institute of Tropical Meteorology, Poona (India) *

(Received January 5, 1976)

ABSTRACT

Gadgil, S., 1976. Time-dependent topographic meandering of a baroclinic current. Dyn. Atmos. Oceans, 1 :127--161

The effects of baroclinic instability of a broad ocean current, flowing in an ocean basin with a plane sloping bottom, on the path of the current are studied. The set of equations governing this path and its variation with depth are the vorticity equation and the heat equation. It is assumed that the vertical and horizontal temperature contrasts are comparable as suggested by observations of the Gulf Stream. When quasi-geostrophy is assumed in addition, this implies that the leading contribution to the heat equation does not contain the vertical advection of the basic stratification. This corresponds to the long-wave approximation of the usual baroclinic-instability problem. The heat equation determines the vertical variation of the path and when this is combined with the vorticity equation, the equation governing the path at one level is obtained. The path equation requires a specification of the direction and curvature at the inlet and these conditions are taken to be time-dependent. When these conditions contain frequencies for which the current is unstable, meanders in the path of the current increase in amplitude downstream of the inlet. When the path at the inlet changes suddenly from one parallel to the isobaths to one making a small angle with them, the region of instability in which the amplitude of the meanders increases, is confined to a restricted segment of the path, at some distance from the inlet. This region becomes advected with the basic current, and its extent increases with time. The amplitude of the meanders in this region increases while their wavelength decreases in time because the shorter waves are unstabler. The increase in amplitude and decrease in wavelength in a restricted segment of the path could lead to eddy formation in a finite-amplitude model and may therefore suggest a mechanism for eddy formation in the Gulf Stream.

INTRODUCTION

The system considered is described in detail in Robinson and Gadgil (1970) (hereafter referred to as TTM) and only a brief description will be given here. We consider the path of an ocean current with no cross-stream variation in velocity in an ocean basin with a plane sloping bottom. The basic assumption is that the change in vorticity produced by the change in depth

* Present address: Center for Theoretical Studies, Indian Institute of Science, Bangalore- 560012, India.

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along the path of the current is balanced by a change in the relative vorticity of the current. Fur thermore, if we allow no horizontal shear, a change in relative vorticity implies a change in the curvature of the path. In fact, it is because this kind of vort ici ty balance can, to a large extent , explain the meanders of the Gulf Stream that we have under taken investigation of broad current models.

In these models, when the direction of the current is along the isobaths at the inlet to the basin, the current experiences no change in vorticity and the path remains parallel to the isobaths. When, however, the path makes an angle with the isobaths at the inlet, there is stretching or shortening of vortex tubes and the path becomes sinusoidal, We study here the transition from a straight path along the isobath to one meandering about the isobath. This simplest, non-trivial example of t ime-dependent topographic meandering was discussed for the barotropic case in TTM. Here we consider the modifications due to baroclinity of the current. They are particularly interesting because of the presence of baroclinic instability. *

We may emphasise, as in TTM, that this model should be considered as one in a hierarchy of models for the path of the Gulf Stream. As such, features of this model will necessarily be present in more realistic complex models. For instance, it can be shown that the equation governing the path of a t ime-dependent baroclinic jet reduces to the one considered here upon linearization (Robinson and Luyten, personal communicat ion) .

MATHEMATICAL MODEL

The ocean basin is taken to have a rigid lid and a plane sloping bot tom. The horizontal axes are chosen so that the depth is independent of x, the downstream coordinate, and decreases uniformly with y, the cross-stream coordinate. The z-axis is taken to be vertical and hence the top and bo t tom axe given in nondimensional coordinates as:

z = 1 and z = 5y

where 6 = b L / H is the ratio of the change in depth over a characteristic distance L and the total depth H. The basin is taken to be bounded on one side by the inlet plane, and infinite in the cross-stream direction.

We neglect dissipation and assume that the motion is hydrostat ic and quasi-geostrophic. When the zonal component of the inlet current U(y,z , t) is taken to be independent of y, the dominant component of velocity along the meridional direction also becomes independent of y if U~ t = 0. We assume this to be satisfied (cf. TTM, eq. 2.3). The equations governing the path of the current in our model are then the vort ici ty equation (1) and the

* The reader is referred to Gadgil (1970) for a discussion of other relevant studies of barocl inic instabi l i ty o f ocean currents such as Tareev (1965) and Orlanski ( I 9 6 9 ) .

NOTATION

Symbol Meaning

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x, y, z coordinates t time

parameter measuring topographic forcing b slope of the bo t tom L horizontal length scale H total depth U velocity parallel to the x-axis U 0 characteristic velocity v velocity parallel to the y-axis f Coriolis parameter Uo/fL Rossby number a parameter mult iplying bo t tom effects k wavenumber c~ frequency c + (¢o) phase velocity c~(c0) group velocity

signal velocity ~__-~u2~- ~ 2 baroclinicity parameter baroclinicity parameter

co c critical frequency p Laplace transform variable

Laplace transform of v R(~,t; ~*,t*) Riemann function h(~,t) boundary function f(x,t) boundary function g(x,~) boundary function S(t) step function

n l ' 2 ~ of the boundary condit ions in 43 and 44 parameters eqs. K )

adiabatic equation (2):

1 f dz [vx t + U(z)vx,,] + ov(x,O,t) = 0 (1) o

vzt + U(z)vzx - - U~v~ = 0 (2)

where v(x,z,t) is the dominant component of the meridional velocity. The parameter 5 is a non<limensional measure of the vertical velocity produced by the sloping bottom and thus also of the topographically induced stretching and shortening of vortex tubes. The rate of change of relative vorticity is of the order of the Rossby number, Uo/fL, where U0 is the characteristic velocity and L is a horizontal scale. Quasi-geostrophy implies that both these parameters are less than unity. The parameter a is the ratio of the parameters 5 and the Rossby number. The meander scale L is chosen such that the change in vorticity produced by the topography is balanced by a change in

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the relative vorticity. In this case, the parameter o becomes of order uni ty and is retained here as a tracer of the bot tom slope.

In the adiabatic equation, we have assumed that horizontal and vertical temperature contrasts are comparable, as suggested by the temperature field associated with the Gulf Stream. This implies that vertical advection of the basic temperature field is unimportant in the 0th order, which consists of horizontal advection balancing the time rate of change of temperature. The reader is referred to TTM for details. In the next section we consider the ele- mentary wave solutions of this set of equations and in the following sections, solve the equations for appropriate boundary conditions.

E L E M E N T A R Y WAVE S O L U T I O N S A N D B A R O C L I N I C I N S T A B I L I T Y

B a r o t r o p i c c a s e

For the barotropic case, the adiabatic equation is trivially satisfied and wave solutions of the vorticity equation of the form exp i ( k x - - c o t ) have the dispersion relation:

k c o - - k 2 + o = O

Thus the wavenumber k, the phase velocity c and the group velocity cg are given for a given frequency co:

co x/co 2 + 40 k±(co) = ~ + - - ~ (3)

c±(co) = 1 2a (4) 20 + co2 + cox/co2 + 40

c~(co) = 1 + 20 (5) 20 + o)2 + ¢ o x / ~ 40

It was found that signal velocity played an important role in determining the nature of the solution of the barotropic problem (TTM, § 4.2). The signal velocity for a given set of inlet conditions can be defined as the group velocity of the dominant frequency in the inlet conditions. In this paper, as in TTM, this dominant frequency is co = 0, which implies in the barotropic limit:

C~gnal = Cg(0) = 2 (6 )

B a r o c l i n i c c a s e

For the baroclinic case it can be shown that the only time-dependent solutions with a separable z-dependence are the wave solutions which are of the form [ U ( z ) - ¢o/k]exp i ( k x - - c o t ) . The wavenumber k and the frequency

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co are related by:

k 3 __ 2 k 2 ¢ o ( U > + k ( ~ o 2 - - o [ ] ) + o ~ = 0 ( 7 )

< US> < US> 

where () and -- denote averages over depth and value at the bottom, respectively. Thus:

1

<¢ - f a (x , z , t )d z -a =- a(x ,y , z = O,t) (8) o

For a given wavenumber k, the frequency co is given by:

¢~ = 1 [~(O -- 2k2(U)) -+ ~/o 2 -- 4k2a~ -- 4k4s 2 ] (9)

where:

a -= -- U s 2 - < uS> -- 2 (I0)

The parameters ~ and s measure the baroclinicity of the basic current U(z), s2is always positive and a positive whenever the current is in the same direc- t ion at all depths and its velocity decreases from top to bot tom.

Baroclinic instability

It can be seen from eq. 9. that the frequency is complex for real wavenumbers whenever:

o ( x / ~ + s2__ ~) (11)

This is the short-wave instability in the long-wave approximation of the usual baroclinic-instability problem (Fjortoft , 1951).

We are, however, interested in the response of a current to t ime<lependent conditions at the inlet x = 0. These inlet conditions contain the contribution of various real frequencies which can be computed by Fourier analysis. Hence it is more appropriate to consider the solution as a superposition of simple waves with real frequencies with their wavenumbers determined by eq. 7. I t can be shown tha t two of the wavenumbers are complex for frequencies higher than a critical frequency. The amplitude of a wave with such a frequency then increases with increasing distance from the inlet as e x p ( J k i l x ) . This is the manner in which baroclinic instability is manifested in the path of the current.

The critical frequency above which two wavenumbers are complex can be computed for a given profile U(z) i.e., for a given set of parameters (U>, (U2), U. These parameters have been determined for:

(1) an exponential profile which decreases from the value uni ty at the surface z = 1 to the value U at the bo t tom z = 0; and

(2) the Gulf Stream profile reported by Fuglister (1965) with the ad-

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di t ional assumpt ion tha t the b o t t o m ve loc i ty is 10% of the surface veloci ty . The paramete rs U, , , a, s for these two profi les t oge the r wi th the critical f requencies and wavenumbers are the first tw o sets given in Table I. No te tha t since the ve loc i ty used for the calculat ion o f the paramete rs and the dep th are t aken to be non-dimensional , the critical f requencies and wavenumbers p resen ted are also non-dimensional .

Since the critical f r e q u e n c y is ob ta ined by solving a cubic equa t ion , an exact express ion in t e rms o f U, , < U2> is cumber some . An a p p r o x i m a t i o n which is good for the first two profi les o f Table I is given by:

2 ~ aUa < U¢> (12) ~c Fj2(3<U2 >_ 42) + 3(2O_ 3<U2>)2

For Rossby numbers of the order of 10 -I, and velocity profiles similar to the Gulf Stream, frequencies higher than about once a month, i.e. periods shorter than a month, are unstable.

Another important feature of these unstable baroclinic waves is the nature of the group velocity. The group velocity for a given wavenumber and frequency is given as:

co 2 -- 4k¢o + 3k2< U2> -- or] cg(k,co) = 2k2 -- o -- 2o9k (13)

For a given real frequency ~9, there are three group velocities corresponding to the three wavenumbers determined by eq. 7. In the unstable regime, two of the group velocities become complex. Furthermore, one or more of the real roots can be negative even in the stable regime. Thus when the group velocity for the dominant frequency is negative, the signal velocity, as defined in eq. 3, becomes negative. In fact, for the inlet conditions chosen in TTM as well as for the baroclinic problem considered here, the dominant frequency is co = 0. The wavenumbers and the group velocities at this frequency are then:

kl(O) = 0 k2,3(0) = + ~.<a~__>

2 U<U2> (14) %1(0)= U Cg2,3(0 ) =2U_<U2>

Note that eq. 14 reduces to eq. 6 in the barotropic limit. Also, eq. 14 indi-

TABLE I

Calculated parameters for three different profiles

Profile No. U <U 2) o~ s kc ~c

1 0.1 0.4 0.2 0.3 0.2 0.86 0.03 2 0.1 0.27 0.14 0.17 0.26 1.06 0.037 3 0.3 0.46 0.24 0.16 0.16 1.11 0.21

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Cg k

Cg ,,

4 0 -

3 0 -

2 0 -

1 0 -

-10

--30 -

- 4 0 -

i i i

0 .4 16 i i I i

c ~ 1.o t~

(a)

5

- 7

- 1 5

-27

-3~

- 4 5

3 7 -

I

17

. /Cg 2

....................... Cg 1

; ' 2 ' o'.4 ' o'.. ' ols '

• - I (hi

- I

I 0 f.~

Fig. 1. The variation of the group velocity cg with frequency co. a. Gulf Stream profile (2 in Table I). b. Profile 3 of Table I. In Fig. la , the real root Cg(~) is plotted, in Fig. lb , the real root ( . . . . . . ) cg 1 and one of the other roots Cg 2 ( ) are plotted, the latter only in the stable region w-here it is real.

cates that cg2, 3 (0) become negative unless:

2 U(U) > ~ U 2) (15)

The condit ion of eq. 15 restricts the vertical variation of the basic profile. However, even when eq. 15 is satisfied, group velocities for other frequencies can be negative. This is shown in Fig. 1 in which variation of group velocity with frequency is shown for the last two profiles in Table I, i.e., the Gulf Stream profile and a profile chosen so as to satisfy eq. 15. The parameters U, (U), ( U 2 ) o f the latter are derived from a profile which decreases linearly from its surface-value unity to its bo t tom value U within a nondimensional depth o f 0.4 and remains U from this point upto the bot tom. Note that this profile is less baroclinic than the other t w o presented in Table I and has higher critical wavenumber and frequency, i.e., is stabler. I t can be seen from Fig. l b that the group velocity for this case is negative between frequencies 0.1 and 0.2. Thus although eq. 15 is significant for the inlet conditions chosen here, it may not have general significance.

DERIVATION OF THE PATH EQUATION

In this section we derive the equation governing the path of a baroclinic current. The downstream component of velocity is given as U(z) and the

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cross-stream component is determined by the adiabatic equation (2) and the vorticity equation (1). Thus the fundamental equations for the baroclinic model are:

Vzt + Uvzx -- U~vx = 0 (16)

1 f [v~t + U(z)vx~]dz + ov(x,O,t) = 0 (17) o

The steady solutions of the adiabatic equation (16) are of the form U(z)~p(x) + J2(z). In the particular case @ = 0, the cross-stream component v(x,z) has the same vertical structure as the downstream component U(z). The form U(z)dp(x) is consistent with the steady form of the vorticity equation (17) provided that the inlet conditions vi = v(0~z, t) and v~ I = v~ (0~z, t) are steady and also have the same z-dependence as U(z). In this case v(x,z) is:

U(X'Z) -~ [ v2+ (V2)uV- VxIj2 7112 c o s Ix ]/__~V(U ")OU __ tan_l \(VXIvI 1/(U2>t~V -u~]J

which is similar to the steady barotropic meander pattern, i.e.:

V(X)barotropie = [V2 + V2i/o] l/2 coS [XV~- - tan-- l (V~q)] (18)

The path of the current, at any level, is determined by the horizontal velocity field (U,v) at that level. The angle between the tangent to the path at any point and the x-axis measures the direction of the path and is given in terms of the velocity field by tan- l (v /U) . The curvature of the path can be expressed in terms of the velocity field as:

Thus the assumption about the vertical structure of v1 and vxi being the same as U(z) implies that the direction and curvature of the path at the inlet are independent of the depth.

As in the barotropic case, we pivot our analysis of the t ime<iependent problem about the steady state and investigate the establishment of steady meander patterns by a suitable choice of inlet conditions. If the time-dependent solutions of the adiabatic equation had a separable vertical structure, the vorticity equation for the baroclinic model (eq. 17) could be reduced to that governing the barotropic meanders by re-scaling the independent variables. However, the t ime-dependent solutions of the adiabatic equation have a separable z-dependence only for simple waves. Hence, for general time-dependent boundary conditions, the constraint imposed by eq. 16 implies a non-trivial vertical structure. This structure determines the general character of the equation governing the meander pattern which turns ou t to be consid- erably different from that governing the barotropic meanders.

In solving the set of integro<lifferential eqs. 16 and 17 we first unravel the

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vertical structure of v by solving the adiabatic equation (16). A complete determination of this solution requires the knowledge ofv at the initial instant t R viz., V(X,Z, tR); at the inlet x = 0, i.e., v(O,z,t) and also at some reference level ZR, i.e., V(X,ZR,t ). If V(X,ZR,t ) is assumed to be known, the solution of the adiabatic equation is determined for given initial and inlet conditions. The search of V(X,ZR, t) is similar to the problem of the determination of a so-called level of no motion in classical oceanography which enables the complete determination of the velocity field from the observed density field via the geostrophic and hydrostatic relations. In general, such a solution does not satisfy the constraint imposed by the conservation of potential vorticity (eq. 17) which determined the path in the barotropic case. Hence, the non-trivial velocity at the reference level has to be chosen so as to satisfy this constraint.

The vertical structure of the solution v(x,z, t) of eq. 16 for arbitrary V(X,ZR,t ) is determined by that of the downstream component U(z) and the initial and inlet conditions. Upon substitution of this solution into eq. 17 the integrals over depth can be evaluated since the z<lependence is known. This yields an equation governing V(X,ZR,t) with coefficients that depend on U and various averages of U(z). This equation can be solved for appropriate inlet and/or initial conditions. Substitution of this V(X, ZR, t) into the original solution of the adiabatic equation yields the solution of eqs. 16 and 17 for the given boundary conditions.

The adiabatic equation

The adiabatic equation (16) is a second~rder partial differential equation in three variables. It is convenient to consider the Laplace transform of v(x,z,t):

~(p,z , t ) -- / e -px v(x ,z , t )dx (19) o

The equation for ~ is obtained from eq. 16 as:

d}zt + pUd}z - - p V ~ = Uv~(Ocz, t) -- U~v(O,z,t) (20)

This is a hyperbolic equation in z and t, with characteristics parallel to the coordinate axes in the z - - t plane. The boundary conditions for eq. 20 can be specified along any open curve in this plane. We take this curve to be a right~angle curve parallel to the coordinate axes (Fig. 2). The characteristics through a point P(p*,z*, t*) intersect this curve at two points, the domain of dependence of P being the curve between these points of intersection. It can be seen that the solution ~ depends upon conditions at the reference level zR, at the initial instant tR, and the cross-stream velocity at the inlet v(0~z,t); the last determining the inhomogeneous part of eq. 20. The reference level zR can be chosen to coincide with the bo t tom z = 0 without loss of generality because,the final solution of eqs. 16 and 17 is independent of the particular reference level chosen. Hence we choose ZR = 0.

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Solu t ion by R iemann ' s me thod . Eq. 20 is solved for arbitrary boundary conditions by Riemann's method. The Riemann function satisfies:

Rz t -- (pUR )~ -- p R U z = 0 (21)

R~(z , t* ; z* , t* ) = 0 (22)

Rt (z*, t ; z* , t* ) = p U ( z * ) R ( z * , t ; z * , t * ) (23)

R ( z * , t * ; z * , t * ) = 1 (24)

This method is particularly useful for this problem because eq. 23 can be in- tegrated at once along the boundary z ~ z. ~, and the knowledge of the Rie- mann function at this boundary simplifies the quest for a solution of eqs. 21--24, which yields:

R ( z , t ; z * , t * ) = [1 + p { U ( z ) -- U(z*)} ]e pu(~)(t- t*> (25)

In terms of the Riemann function, the solution of eq. 25 for boundary conditions along B O A (Fig. 2) is given as (Courant and Hilbert, 1953, p. 454):

A

~(p) = ~ [ ¢2(A)R(A) + ~ (B)R(B) ] - - ~ f d t [ R t ~ - - ¢2tR -- 2pURe2] o

B

-- ~ f dz [Rz ¢2 - - R ~z ] + f f Rh( z , t ) d zd t (26) o G

where G is the area OAPB and h is given as:

h(z , t ) = Uvz (O,z,t) - - Uzv(O,z, t)

/ / / /

O(P,ZR,t R) Atp,Z~, ) Z

(0 ,0 ) s (x ' -~ t ;o ,o ) " x

Fig. 2. Character is t ics and domain o f d e p e n d e n c e for t~(p,z *, t*) in the z - - t plane.

Fig. 3. Character is t ics and doma in o f d e p e n d e n c e for P ( x *,z *, t *) where x * > U(z *)t*. The line x = U(~)t is a character is t ic on any plane z = ~. The line PA is parallel to the character is t ic in the plane z = z *. The po in t C is the pro jec t ion o f P on the b o t t o m and B C is parallel to the character is t ic x = Ut on the b o t t o m . The so lu t ion at P depends u p o n data along A B and BC.

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Substi tut ion of eq. 25 into eq. 26 and inversion of the transform yields the solution of the adiabatic equation in terms of the boundary conditions. Let:

v(x,O,t) = f ( x , t )

v(x,z , t R) = g(x ,z ) (27)

At this point we take the initial instant tR to coincide with t = 0 to facilitate the description of the domains of dependence for various cases. We will, however, choose a different ta while solving this problem for specific initial and inlet conditions.

In terms of the functions f and g the solution for tR = 0 is:

- - (o v(x , z , t ) = f (x , t ) - f ( x -- Ut, O)S(x - Ut) + f , t - - S t - - t

+ ( u - ~ f f=[x - U(t - $ ) , $ l S [ x - U(t- ~)1 d~ +g(x -- U t , z )S (x - -Ut ) 0

+ (x

U + f dkS(x - - k t ) [2tg= (x - - Xt, k) -- (X -- U)t2gxx(X - - ~ t ,~ . ) ]

U

d~ t - x / U l D x

where S is the step funct ion defined as:

S ( t ) = 0 t < 0

S(t) = 1 t > O

(28)

t

• l / / X:Ut, Z:O

z ~ X :U (Z*) t, Z:Z

~./~(6.o. "x 6x90)

Fig. 4. Charac te r i s t i c s a n d d o m a i n o f d e p e n d e n c e for P(x *,z *, t* ) whe re x * < Ut*. The l ine t h r o u g h P paral lel t o t he charac te r i s t i c in t he p l ane z = z* m e e t s the l ine x = 0 a t D. T h e p r o j e c t i o n o f P o n t he b o t t o m is C a n d BC is again paral lel t o t he charac te r i s t i c o n t h e b o t t o m . T h e l ines DB and BC give t he d o m a i n of dependence .

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/)¢t / / X:Ut, Z:O

7./ ' . ccx;o,¢~

" -~':x - ~ o , o ) • x

Fig. 5. Characteristics and domain of dependence for P(x *,z *, t*) when x * < U(z *) t*; x * > Ut*. Here the line through C parallel to the characteristic on the bottom meets the x-axis in B while the line through P parallel to the characteristic in z = z* meets the line x = 0 in D. The point E is in the plane x* = U(z)t*. in which the projection of P lies on the characteristic. The domain of dependence is DE, EB and BC.

F r o m the expression (28) it can be seen tha t at any level z, the line x = U(z)t is a characterist ic in tha t the na ture o f the data p icked up by the solu- t ion is d i f ferent on di f ferent sides o f this line.

If the po in t P(x* , z* , t * ) at which the solut ion is required lies be tween the characterist ic x = U(z*) t in the plane z = z* and the line t = 0 in tha t plane, then x * > U(z*)t*. If we assume tha t U(z) is a m o n o t o n i c a l l y increasing func t ion of height, then the p ro jec t ion o f P on the b o t t o m z = 0 necessarily lies be tween the characterist ic x = Ut and the line t = 0 on the b o t t o m . This case is shown in Fig. 3 in which U(z) is assumed to be linear for purposes o f i l lustration. The solut ion in this case depends u p o n the initial cond i t ion o f the veloci ty along the line A B in the plane t = 0, and the veloci ty at the b o t t o m (along the line BC in the plane z = 0).

On the o the r hand, if the p ro jec t ion o f the po in t P on the b o t t o m lies between the characterist ic x = Ut and the line x = 0, x* < Ut* and for m o n o - tonical ly increasing U(z) it fol lows tha t x * < U(z*)t*. In this case (Fig. 4), the solut ion at P depends u p o n the veloci ty at the inlet x = 0, along the line BD and the veloci ty at the b o t t o m along the line BC.

In the in termedia te case x * < U(z*) t* and x* > Ut*, the solut ion depends u p o n bo th initial and inlet da ta toge ther with the veloci ty at the reference level a long BC (Fig. 5). We no te tha t initial da ta is p icked up only up to the level at which the po in t lies on the characteris t ic x *= U(z) t* .

The pa th equat ion

The equa t ion governing the veloci ty at the b o t t o m f ( x , t ) can now be ob- ta ined by subs t i tu t ion o f eq. 28 if ta is t aken as zero, or a modi f ied fo rm of eq. 28 if ta differs f r o m zero into eq. 17 and evaluat ion of the integrals over the dep th . I t should be no t ed tha t the general charac ter o f this equa t ion will no t be al tered by the initial cond i t i on g(x , z ) and the inlet da ta h(z , t ) since

139

bo th these will contr ibute only to the inhomogeneous driving terms. In addition, the steady solutions of eqs. 16 and 17 correspond to inlet condit ions which have the same vertical structure as U(z) , for which:

h ( z , t ) = 0 (29)

We restrict the t ime~lependent problems to those with inlet conditions satisfying eq. 29. This is equivalent to assuming that the t ime<lependent direction and curvature at the inlet are independent of depth. In order to s tudy the problem of determination of the velocity at the reference level in its simplest form, we assume also that there is no cross-stream velocity at the initial instant, i.e.:

v(x , z , t i t ) = g ( x , z ) = 0 (30)

Under the condit ions of eqs. 29 and 30 the expression (28) for an arbitrary tit becomes:

v ( x , z , t ) = f ( x , t ) + (~- -~ U 1 f( O,t - - x / D')S( t - - tit - - x / £r) \ t J /

t

+ ( u - f r . ( x - ot + o ,e) s(x - vt + O~)d~ (31) tit

Operating on eq. 31 with a / a t + U a / a x , we get:

vt + Uvx = [t + Ufx (32)

If eq. 17 is also operated on by a / a t + u a / a x eq. 32 can be used to eliminate v and obtain an equation for f alone, after the evaluation of the integrals over the depth. This gives:

f~tt + 2(U>fx~t + <U2>f~x + o( f t + Uf~) = 0 (33)

ELLIPTIC NATURE OF THE PATH EQUATION AND ITS CONSEQUENCES

The path equation for the baroclinic case is given by eq. 33. This equation has characteristics given by t = const., x - - (U) t + ist = const.

Two of the characteristics are seen to be complex whenever the baroclinicity parameter s is non-zero. The equation governing the path of a baroclinic current is therefore, elliptic. Thus baroclinicity modifies the basic nature of the equation.

Ellipticity makes the present problem an unusual one, in view of the fact that one of the independent variables is the t ime t. A so-called well-posed problem for an elliptic equation is one in which the boundary condit ions are specified on a closed curve, e.g., Dirichlet and Neumann problems for Lap- lace's equation. For such a problem, the solution at any point depends upon the condit ions at all points on the boundary. This implies that if the bound-

a r y condit ions were specified on a closed curve in the x - - t plane for the

140

present problem, the solution would depend upon conditions corresponding to a future time, which is clearly unphysical.

The other alternative is to treat the problem as a Cauchy problem in which the function and its normal derivatives are specified on an open curve in the x-- t plane. The solution for an elliptic equation with Cauchy data is known to be mathematically unstable in the sense that a small change in the boundary conditions can produce a large change in the solution at a large distance from the boundary. The Cauchy problem for an elliptic equation is considered to be improperly posed because of this instability. Such problems are not, however, necessarily unphysical. G.I. Taylor's analysis of the instability of a liquid surface accelerated normal to itself gave rise to an initial value problem for Laplace's equation which is also an improperly posed problem (Courant and Hilbert, 1962, p. 230). The mathematical instability of that problem was a manifestation of the physical instability of the system.

It was shown in an earlier section ("Mathematical model") that the amplitude of a wave of a frequency for which the current is unstable increases with increasing distance from the inlet as exp( Ik i i x ) . This behaviour is precisely that of a mathematically unstable solution of a Cauchy problem of an elliptic equation and is seen to arise from the inherent instability of the baroclinic current. The amplitude of the solution will grow in certain regions of the x-- t plane, whenever the inlet conditions have energy in the frequencies ¢o > ¢o c. In particular, for the special case of a fiat bo t tom this holds for all non-zero frequencies.

COMPARISON OF BAROCLINIC AND BAROTROPIC PATH EQUATIONS

In the barotropic case, the equation governing the path of the current is the vorticity equation:

v~t + v~ + ov = 0 (34)

where the downstream velocity U has been assumed to be unity. Eq. 33 governing the velocity at the reference level f(x, t) is a third-order partial differential equation in x and t, whereas eq. 34 for the barotropic velocity is of the second order. This is because the process of elimination of v(x,z, t) from the expression (31 ) and the vorticity equation (17) involved differentiation with respect to x and t. However, eq. 33 for f (x , t ) does not require more boundary conditions than the barotropic equation (34) since it can be shown that f~x(O,t) is related to f(O,t) and f~{0,t). Integration of eq. 17 in x yields:

, y f (vt + Uvx )dz + o v(x,O,t)dx' = F(t) o o

where:

(35)

1

F(t) = f [vt(0~z,t) + U(z)vx (0~z,t)] dz 0

(36)

141

Substituting for v(O,z,t) in terms of f f rom eq. 31 we get:

F(t) = Aft(O,t) + Six (0,t)

where:

A = 2(U) U-- (US) and B = ~ (37)

Now, operating on eq. 35 with a/a t + Ua/ax and combining with eq. 32 we get, upon evaluation of the expression at x = 0:

_ oUf(O,t) fxx(O,t) A -- 1 f t t(O,t) + B -- 2(U) fxt(O,t) (38) (US) (US) (US)

The transition from the baroclinic case to the barotropic case is seen to be smooth since in the barotropic limit -- (U) = (US) = U = 1 -- eq. 33 reduces t o :

(f~t + fxx + of) = 0 (39)

SOLUTION OF THE PATH EQUATION

Since the path equation 33 is of the second order in t ime and of the third order in space, it is solved by using Laplace transforms in space. We define:

~(p,t) = f e -px f (x , t )dx o

So that:

1 ~/+i~ f(x, t) = ~-~ f ex ~(p,t)dp

On transforming, eq. 33 becomes:

tl/tt +(2 (U)p +p) ~t + (p2(US)+ a~r)~ = R(p,t) (40)

The driving term R(p, t) depends upon the conditions specified at the inlet and is given by:

R(p,t) f .Aftt + Bfxt f l = + 2(U)ft + (US)f:, +p(US) P ~x = o

The solution to eq. 40 is:

I~(p , t ) = C ( p , t ) e m l t + D ( p , t ) e m 2 t

m l , 2 = - - ( U ) + 0 + + o o t - - p 2 8 2

(41)

(42)

142

The coefficients C and D depend on R ( p , t ) and hence on the inlet conditions. It can be seen that for large p :

Lim rnl, 2 = --p( -+ is) p ~

Since the inversion contour in the p plane extends from --~ to o~ for imaginary p, the function @(p,t) will become infinite on this contour unless C and D decay more rapidly then exp(+isp). Thus for arbitrary inlet conditions, the transform @(p,t) does not exist. Note that the presence of s in this exponential implies that this restriction is necessary only for the baroclinic case and is a consequence of the mathematical instability discussed earlier ("Elliptic nature , . ."). Choosing the inlet conditions such that the transform exists, we take:

V [1 + erf(tx/~)] (43) f ( O , t ) =

fx(O,t) = --Kft(O,t) (44)

where:

2 f e -~'2 d~ = 1 -- erfc(z) erf(z) ~ 0

Note that the inlet condition is consistent with the initial condition (eq. 30) provided that the initial instant ta is chosen as --~. Henceforth we assume tR = - -~o .

The direction and curvature at the inlet is changed in time of order 1/~/~, ~ being arbitrary. In the limit of infinite fl, eqs. 43 and 44 imply a sudden change in angle and an input of impulse of vorticity, the inlet conditions considered for the barotropic case discussed in TTM. Here also, we will be mainly interested in the limit of infinite ~ for all the cases discussed. For the conditions of eqs. 43 and 44 the solution to eq. 40 is:

( A - - K B ) ( m l I 1 - - m212) ~(p,t) - <U2> erfc(--tV~) +

m l m 2 p ( m l - - m2)

(2 -- K)(11 -- 12) + . . . . . . . . . . . . . . . . +

m I - - m 2

where:

p< U2> (m211 - - m112)

ml rr/2(m I - - /?12 ) (45)

-~ U (-- (46) Ii 2 ~exp(ml,2t + m12 2/4~) erfc t ~ -m1'2~ • v g !

Definitions of A and B are given in eq. 37. Inversion of eqs. 45 and 46 yields the solution to eqs. 33, 43 and 44.

In order to gain insight into the relation between various features of the time-dependent path and the baroclinicity of the current, we will consider the

143

inversion of eqs. 45 and 46 first for the simplest case, i.e., barotropic flow on a flat bot tom, and then baroclinic flow on a flat bot tom. Meandering paths over topography are then discussed, first for a barotropic current and then for a baroclinic current. Although the expression (45) can be inverted exactly for barotropic and baroclinic flow on a flat bot tom, and the solution for barotropic flow on a sloping bo t tom has been discussed extensively in TTM, approximate expressions for all these cases are derived in the Appendix by using the saddle-point method. This provides the necessary background for deriving an approximation to the solution for baroclinic flow over topography which contains features of all the other cases.

Barotropic f l o w on a f lat b o t t o m

In this case eqs. 45 and 46 reduce to:

l e r f c ( - - t x / - ~ ) - - K e x p ( - - p t + p 2 / 4 ~ ) e r f c ( - - t @ + 2 - ~ ) ~ (p , t ) = ~-~ ~-p

This implies:

Lim f ( x , t ) = (1 - - K ) S ( t ) + K S ( t - - x )

(47)

(48)

o r :

Lira f ( x , t ) = S ( t ) - - K S ( x - - t)S(t) (49)

Thus ,when the direction at the inlet is changed instantaneously, the current starts flowing in a direction making an angle tan -1 (1 -- K) with the initial direction, immediately after the instant t = 0. At any position x, the steady state, which in this case is a straight path, is achieved after the instant t = x by means of another discontinuity of magnitude K in the velocity v across the line x = t.

The main feature of the solution is then discontinuities 1 -- K and K in velocity v across the two characteristics t = 0 and x = t emanating from the point (x = 0, t -- 0) at which inlet conditions are changed. In our model, gravity waves have been filtered. The infinite speed with which the discontinuities propagate in our model will be replaced by the speed of the gravity waves in a model in which they are retained. Gravity waves can transmit pressure jumps, but since the discontinuities in our solution are in the velocity field itself, it may be more reasonable physically to choose K so that the discontinuity across t = 0 vanishes. Henceforth, in all cases considered we choose K to satisfy this condition. It is shown in the Appendix (A1) that an approximate solution of this case, obtained by inversion of eq. 47 by using the saddle-point method, has the important feature discussed above.

144

Baroc l in ic f l o w on a f la t b o t t o m

Consider now the path of a baroclinic current on a flat bot tom. In this case eqs. 42 and 45 become:

ml, 2 = --p( T- is}

2-U erfc(--tx/~) + (A - - K B ) ( m l I 1 - - m212) qJ(p,t) 2isp 2

(2 -- K<U2>)(I1 - - 12) (m211 - - m112) + + 2isp 2isp 2

(50)

This can be inverted exactly to yield:

f ( x , t) = U (A - - K B ) [ 1 + erf(tx/~)]

+ - (1 + erf[x,'~{t --x(<g> --is)/<U2>} ])

+ 1 (1 + e r f [ @ { t - x( + is)/< U2>} 1)

P1 -= (1 - - A + K B ) U and P2 -= K U ( U 2 ) P1 2 2s 2s

(51)

Discuss ion o f the so lu t i ons on a f la t b o t t o m . Consider first the solution (eq. 51) for large ~, i.e., when the change in the inlet angle is accomplished in a short time. Since the arguments of the error functions occurring in eq. 51 increase with increasing ~, we have to consider the behaviour of error functions of complex arguments for large arguments in order to explore the nature of the solution for large ~. In this limit (Albromowitz and Stegun, 1965, p. 298):

Lim erf z = 1 if larg z l < ~/4 (52)

and the asymptotic form for large z is given as:

Lim e r f z = l - - - - Z- - -~ ~

e-Z2 I1 + (-1)" ~Vl f f Z rn = l

1 • 3 . . . . - (2m - 1)1 / (2z

For the particular error functions of complex arguments in eq. 51 the con-

for:

largzl < 3~/4 (53)

145

dition of arg z in eq. 52 becomes:

t ( U ~ ) _ x < 1 (54)

The region defined by eq. 54 lies below the line x = t( U2)/(< U) - - s ) and above the line x = t{ U~>/{< U) + s) in the x - - t plane, the region between these two lines is the hatched region in Fig. 6. Except in that region then, the solution is bounded in the limit of infinite ~ and the error functions reduce to step functions. The solution is given as:

Lim f ( x , t ) = 0 : t < 0

= U--(A - - K B ) : x ( ( U > - - s ) > t > 0 (55) 

= U : t > (U)+--------~Sx ( U'z>

For finite ~, the funct ion is, of course, continuous. Comparison of the baroclinic solution (eq. 55) in these regions with the

barotropic solution (eq. 49) shows that the former differs only in the occur- rence of the baroclinicity factors A and B. Also, there is no jump across t = 0 when we choose:

A = K- B (56)

In the region of the x - - t plane which does not satisfy expression 54 use of expression 53 yields an approximate expression for the solution:

U(A - - K B ) [ 1 + er f ( tv~)] + (1 - - A + K B ) U " f(x,t)

exp F_ I(t_ s'=' l l . L (\ <U')'/ i-U-~)J

(---~-~]---~ s in {~) \ -- ~-~,)j (57)

t<uZ>

x" ~u-'ff~- s

Fig. 6. The region of instability in the x - - t plane is between the lines x = t(U2)/((U> - - s) and x = t(U2)/((U) + s).

146

It can be seen from eq. 57 that the amplitude of the solution depends upon:

l_ L( U 2) 2 ( U27

In this region, s2x 2 > ( t ~ U 2) - - x ( Ui )2 (from eq. 54) and hence the amplitude of the solution increases with distance from the lines t = x(/32)/((U) + s). This region can, therefore, be called the region of instability. The extent of this region in x increases with time and so does the maximum amplitude. The sine and cosine terms in eq. 57 show that the solution oscillates in this region, the wavelength of the oscillations for a given time, decreases with increasing time and with increasing distance from the inlet. This is illustrated in Fig. 7 which shows the time evolution of the solution for fl = 100 for a particular baroclinic profile U(z).

These features of the solution in the region of instability are, however, not restricted to large values of ~. The expression (57) is a good approximation to the solution whenever the arguments of the error function are large. Thus when ~ is small these features will occur at sufficiently large distance from the lines t = x{ U2)/(~ U) + s).

t (u ~) / " x=t/3~ (u~+s

/ /

/

//

j / J

x:o '

Fig. 7. Time evo lu t ion of the pa ths on a f la t b o t t o m f rom eq. 51. At the given t imes t = 0.1, 0.2, 0.3, f(x,t) is p l o t t e d versus X. The prof i le U(z) chosen is the f irst one in Table I. The l i nesx = t and x = t/3 r ep re sen t the b o u n d s of the region o f ins tabi l i ty . The t ime scale 1/x/~ is t a k e n to be 0.1.

147

Constraint on allowable U(z). There is another feature of this solution which should be noted. It is clear from Fig. 6, that for an instantaneous change in inlet conditions at t = 0, the region of instability occurs after the instant t = 0 only if:

( U ) > s

S -- ~/< V2> - - ( U ) 2

i.e.:

2 ( U ) 2 ) ( U 2)

Thus the results are physically reasonable only ff the basic profile U(z) satisfies eq. 58. This implies a restriction on the baroclinicity of the allowable class of profiles. Thus for a profile in which the velocity varies linearly over a quarter of the depth and then remains the same to the bot tom, eq. 58 requires the bo t tom velocity to be greater than 10% of the surface velocity.

In terms of group and phase velocities, which are complex for this case, eq. 58 implies that the magnitude of their real part is greater than the imaginary part. We may note tha t this constraint is not too stringent and in fact, is satisfied by the Gulf Stream profile reported by Fugiister (1965). The asymptotic approximation of this solution is discussed in the Appendix (h2).

(58)

BAROTROPIC FLOW ON A SLOPING BOTTOM

We discuss here the major features of the solution for a path of a barotropic current on a sloping bo t tom discussed in TTM. The results will be re- derived in the Appendix (A3) by using asymptotic approximations in a framework which can be easily extended to the baroclinic case.

Consider the case ~ -~ ~ corresponding to an instantaneous change in inlet conditions at t = 0. The discontinuity in the inlet conditions at t = 0 is car- ried along both the characteristics t = 0 and x = t through this point (x = 0, t = 0). There is a jump of magnitude 1 -- K across the characteristic t = 0 and another of magnitude K across the characteristic x = t in the meridional velocity v. This behaviour is, in fact, not dependent on the topography and is present also in the flow of a barotropic current over a fiat bot tom, subject to these inlet conditions.

Shortly after the jump across t = 0, the solution has a number of widely spaced zeroes at large distances downstream of the inlet. As time progresses, these zeroes move in towards the inlet. The local wavelength of these retrograde waves is large compared to the final steady state.

The solution changes character across the line x = 2t. In the region between x = 2t and the characteristic x .= t which delimits the transient solu- t ion, the solution markedly resembles the final steady state. This steady state is achieved for t ~ x.

The behaviour of this solution was interpreted in TTM in the following

148

manner . The na tu re o f the so lu t ion for large x/ t depends on tha t o f the waves o f large f requencies co which have the highest phase and group velocities, i.e., the wave f ronts . The curva ture K de te rmines the energy in these waves. The d o m i n a n t c o n t r i b u t i o n to the inlet condi t ions chosen here comes f rom waves o f f r e q u e n c y co = 0. The "s ignal" travels wi th the group veloci ty cor responding to this d o m i n a n t f r equency cg(0), which in this case equals 2. This p ropaga t ion o f the signal along x = 2t is mani fes ted in the marked resemblance o f the so lu t ion to the final s teady state be tween the regions x = 2t and x = t.

Discussion of the solution for baroclinic flow on a sloping bottom

The so lu t ion fo r the pa th o f a barocl inic cur ren t on a sloping b o t t o m can- no t be ob ta ined exact ly . Hence we use the insight gained f ro m the simple p rob lems discussed in previous sect ions to ob ta in some fea tures of the solu- t ion. The m e t h o d o f saddle po in ts is again e m p l o y e d in the inversion o f eqs. 45 and 46. All the details are p resen ted in the Appendix . Here, on ly the results are discussed.

The discont inui t ies across the character is t ics in the limit/~ -+ o~ are identical to those for the solut ion o f the f low of a barocl inic cur ren t over a f lat b o t t o m , since t hey do no t depend u p o n the t o p o g r a p h y . Again the curva- tu re K is chosen so as to make the so lu t ion con t inuous across t = 0, bu t none of the o the r results depend on this par t icular choice of K.

As in the case o f a barocl inic f low over a f lat b o t t o m , the solut ion is unstable in a region be tween the lines t = x ( U2)/((U) +- s). In this region, the ampl i tude o f the oscillating so lu t ion increases and the wavelength of the oscil lations decreases with increasing dis tance f ro m the inlet. At a given posi- t ion , the wavelength o f the oscil lat ions decreases wi th t ime. This is to be ex- pec ted since the instabi l i ty in our mode l is a short-wave instabil i ty.

Again, there is a very i m p o r t a n t res t r ic t ion on the zonal c o m p o n e n t of the inlet ve loc i ty discussed in the sect ion: "Cons t r a in t on al lowable U(z)" (of eq. 58), viz. :

22 > <U2> (59)

It is necessary tha t this cond i t ion is satisfied in order to ensure tha t a change in the inlet condi t ions at t = 0 does no t imply a change be fo re the ins tant t = 0 at some dis tance f r o m the inlet.

The na tu re of the so lu t ion immedia te ly a f te r the ins tant t = 0 is similar to tha t of the ba ro t rop ic case in tha t the pa th consists o f long waves whose wavelength increases wi th increasing dis tance f r o m the inlet. This wavelength decreases wi th t ime and meanders are f o r m e d by zeroes moving in f rom large x.

The o the r i m p o r t a n t fea ture of the ba ro t rop ic meander ing cur ren t , viz., the p ropaga t ion o f the signal wi th group ve loc i ty cor responding to the domi- nan t f r e q u e n c y is present fo r the barocl inic case on ly w h en an addi t ional

149

condit ion is satisfied by the zonal componen t of the inlet current viz.:

cg(0) > 0

i.e.:

2U(U) > (U2) (60)

In this case the signal propagates along x = cg 2 s(0)t (of eq. 5) and from that t ime onwards the solution contains a term wh{ch describes the final steady state.

The situation is quite different when (59) is not satisfied, i.e. Cg 2 s(0) are negative. This case is reminiscent of the propagation of electrom'~net ic waves in the presence of absorption. In a region of absorption, the group velocity of these waves can become negative and in fact in this region the signal becomes rather blurred, a slow change replacing the sudden transition obtained in the absence of absorption (Brillouin, 1960). For the problem discussed here two possibilities exist. Either the phenomenon of propagation of signal is absent, or the phenomenon exists, bu t the signal velocity is different f rom the group velocity at the dominant frequency. Which of these is true has to be determined by numerical evaluation of the inversion integrals, which is beyond the scope of the present paper.

Sorting out the other features of the f low of a baroclinic current also requires numerical evaluation of some integrals, which will not be done here. Thus, although we have deduced some features of the baroclinic solution, all the details cannot be obtained wi thout numerical integrations. It does not seem worth-while to undertake this for the simple linear model discussed here. The results obtained above are presented in the hope that they will be useful in the interpretation of the solution of a more realistic non-linear model.

Solution of the adiabatic equation

Finally we may ment ion that once the solution for the path of a baroclinic current at z = 0 is obtained, the path at all depths can be obtained by combining with eq. 31. However, since we have assumed that eqs. 29 and 30 are satisfied, no dependence is introduced by the initial or inlet conditions and the vertical variation is determined entirely by the basic profile U(z). In fact, the solution is of the form ~(x,t) + U(z)¢(x,t), the x--t dependence being similar to that o f the path of the bo t tom. Formally the path of a baroclinic current U(z) on a sloping b o t t o m at all levels can be obtained by using eq. 31 for general initial and inlet condit ions by the methods used here.

IMPLICATIONS FOR THE GULF STREAM

If the present model incorporates some of the essential processes of the Gulf Stream meanders than one would expect the stream to be baroclinicaUy

1 5 0

unstable in the sense discussed here. An interesting feature of the baroclinic problem discussed here is the well-defined region of instability in the x - t plane. In this region, the amplitude of the cross-stream velocity becomes much larger than that outside and furthermore the path of the current oscillates rapidly. This behaviour is reminiscent of the observed elongation of a meander, in a small segment of the Gulf Stream, before the formation and breaking off of an eddy (Fuglister and Worthington, 1951). This suggests that a mechanism for eddy formation is inherent in this baroclinic model. However, at this stage this is a speculation and a more detailed analysis of a non-linear model will have to be done in order to substantiate the claim.

A C K N O W L E D G E M E N T S

I am grateful to Professors D.J. Baker, A.R. Robinson, P.H. Stone and Dr. J.R. Luy ton for helpful discussions, and the referees for useful com- ments. A part of this research was supported by office of Naval Research under contract No. 014-67-A-0298-0011 to Harvard University. It was com- pleted when I was a Pool Officer of the Council of Scientific and Industrial Research attached to the Indian Institute of Tropical Meteorology.

A P P E N D I X

We derive approximate expressions for the solutions of barotropic flow over a flat bo t tom, baroclinic flow over a flat bo t tom and barotropic and baroclinic flows over a sloping bot tom, using the saddle-point method.

Note that the saddle point for a general integral I of the form:

~. "y+ i~

I = ~ ; exp[xh(p)] g(p)dp (A1) 7 - - i~o

is given by p =/~ where:

d__h = 0 (A2) dp p=~

If the path of integration can be deformed so as to coincide with the path of steepest descent (eq. A3) through the saddle point i.e. :

Im h{p) = Im h ~ ) (A3)

then the contr ibut ion of the saddle point is given by (Carrier et al., 1966):

g(~) exp[xh~5)]

2~ x/-xl d2h/dp 2 Ip =~,

For large x then (A4) is an approximate expression for I.

(A4)

151

B a r o t r o p i c f l o w o v e r a f la t b o t t o m

Consider, now, the approximat ion to the solut ion of barotropic flow over a flat b o t t o m obtained by using the saddle-point method . The first term in eq. 47 can be inverted to yield ~ erfc(--tV~), which describes the final steady-state solut ion in the l im~ f3 -* o o . The second term, then describes the transient. We define:

7--ioo

The integral 11 is now evaluated by the saddle-point method . The integrand has a pole at p -- 0 with residue:

R = - - I erfc(--tx/~) (A6)

In this case the saddle poin t ~ is given by:

-- --2fi (x -- t)

The path of steepest descent is parallel to the imaginary axis in the plane. We evaluate I by changing the pa th of integration to this path and using (A4). The cont r ibut ion o f ~ is then:

I erfc(--x/-flx) exp [--fi(x -- t) 2]/21rx/~(x - - t)

For large ~3, the cont r ibu t ion of the saddle poin t is thus restricted to the ne ighbourhood of the characteristic x -- t. We npte tha t the pole p = 0, lies between the original pa th and the path of steepest descent wherever x > t and hence its residue contr ibutes only when x > t. Thus the approximate solut ion is given by:

f ( x , t ) = ~[1 - - g S ( x - - t)] erfc(--tx/~) + K erfc(--x/--flx) e_~(x- t)2 2 2 V ~ ( x -- t)

L i m f ( x , t ) = S(t)[1 - - K S ( x - - t)]

(A7)

(A8)

Baroc l in ic f l o w o n a f la t b o t t o m

The solut ion describing the path of a baroclinic current on a flat bo t tom, can be expressed as:

- - J 1 = I erfc(--tx/-~) + ~ [--(A --KB)(( U> - - i s ) + 2( U> --K f ( x , t )

J2 --( +/s)] + ~-~ [(A - -KB) ( - - i s ) --2<U) + K<U 2) + (U) - - i s ] (A9)

152

J1,2- 2 2rri f dpp {exp[p(x --t +- ist) + ( • is)2p2/4{J] •

e r fc ( - - tV~- -p ((U> ~-is)] t 2-x/~ ]~ (A10)

On assuming that there is no velocity discontinuity for large ~ across t = 0, i.e. eq. 56, expression (A9) reduces to:

f ( x , t ) : ~e r f c ( - - t x /~ )+J1 F(U 2> is~ J2 I<~ . is)] ~ s L - ~ - - ( + - ~ ( -

(All)

Note that the integrals Ji,2 have a pole at p = 0. The residue of this pole is (U--/2) erfc(--t~) when it contributes to both J1, and J2.

The saddle points of J1,2 are given by:

-2~ p = ~-~ [ (x(( U> 2 -- s 2 ) -- < U2>t} + is(2x -- t< U2>)] (AI2)

It can be shown that the path of steepest descent through the saddle point for J1 intersects the imaginary p axis below the origin only when x > t< U2>/ and in that case the saddle point lies in the lower left quadrant of the p plane. The pole lies between the original path and the path of steepest descent when x > t< U2>/ for both Jx and J2 and hence the contribution of the pole to the integrals is given by:

erfc(--t~/~)S (t <U2>~ . - - x ~ > ] (AI3)

Consider now the contribution of the saddle points. Once the path of integration coincides with that of steepest descent through the saddle points, the contribution of the saddle points can be evaluated by using (A4). Combining this with (AI3) we get an approximate expression for:

f(x,t) ~ g - - x < U¢>t~

[ + i s ) ] ~ + exp [--~(x/(( U> -- is) - -0 2 ] (x )

4 i sx /~ -- is t

[ / U - ( - i s ) ] + exp{--fl[x/( + is) --t] 2 } (A14)

It is clear that the region of instability and the behaviour of the solution in this region is well described by the above expression. Furthermore it can be shown that (A14) reduces to eq. 55 in the limit of infinite/3, for the particu-

153

lar value of K chosen. Thus this approximation describes all the features of interest.

Barotropic flow on a sloping bottom

For barotropic f low over a sloping b o t t o m using eq. 56 to determine K, we get:

erfc(--tx/~) pI ~(p,t) = p -- (A15)

2 ( p 2 + a ) p 2 + a

O 2

The first term of the right part of eq. A15 can be inverted exactly to yield:

1_ cos (xv~) e r fc ( - - tv~) (A17) 2

Thus in the limit of infinite/3 this term describes the final steady state after t = 0. The transient is given by:

1 7+i~' J=--47r--~ f pdp exp[px - - t ( p + p ) + (p+p)2 4~]-

7- - i~ p2 + (7

erfc [ - - t v ~ +P + o/p~ (A18) k !

The integral J has poles at p = -+iv/-E and the sum of the residues of the poles is seen to be:

R = _ 1 erfc(--tx/~) cos(xx/a) (A19) 2

The residue cancels exactly the term (A17). Hence in the limit of infinite the solution would contain a term describing the final steady state only when the poles do not contr ibute to the integral J.

The integral J has saddle points at:

- -2 f l (x - - t ) ; o . + i V-x at (A20) 2~t ' - -- t

The original path of integration is taken to the right of the poles and the last two saddle points, i.e., 7 > V~. For convergence at large p, when t > x the path can be changed to one lying entirely in the right half plane (Re p > 0) and coinciding with the steepest descent path of the first saddle point in its neighbourhood. In this case neither the poles nor the other saddle points contr ibute. The contr ibut ion of the first saddle point can be evaluated by (A4) and is proport ional to:

1 exp[--~(x -- t) 2 ] Vr~ ( x _ t)

154

Thus in the limit fi -+ oo this contribution vanishes except along the characteristic x = t and the final steady state is achieved for x < t. This is consistent with the analysis of characteristics discussed in TTM.

The adjustment from the initial flow along the isobars to the final meandering path occurs in the region x > t > 0 of the x - - t plane. For x > t > 0, all the saddle points contribute to the integral. The integration path is deformed so as to coincide with the steepest descent path iP the vicinity of each of the saddle points. Again, the contribution of the first saddle point vanishes in the limit of infinite fi except along the characteristic x = t. The contribution of the second is proportional to ~-5/2 t-2 e-~t2, which also vanishes for large ~ except along the characteristic t = 0. Thus, only the last two saddle points, which for this case lie on the imaginary axis, are significant for inlet conditions which change instantaneously.

For x >> t, these saddle points lie close to the origin and the poles lie between the original path and the path of steepest descent through these saddle points (see Fig. 8c). Thus the poles contribute to the solution thereby cancelling the steady-state term obtained by the inversion of the first term in eq. 59. The asymptotics for small t / x can be obtained by using (A4) to evaluate the contribution of these saddle points, and is given by:

t314 . . . . . . . . e x p [ + 2 i x / t ( x - - t ) +- i 7 r / 4 ]

( x - - t ) l / 4 ( x - - 20

Thus initially, the solution consists of long waves whose wavelength increases with x and decreases with t. At x = 2t, the saddle points coincide with the poles and from then on the poles do not contribute to the integral (Fig. 8b,a). Hence, the solution contains a term describing the final steady state for x < 2t and we may define the arrival of the signal to occur at the instant t =

Imp C (a) !

.... ~ __~L~

Imp Imp

!

1

i

Rep

Fig. 8. Posi t ion o f saddle points, @, and poles, × , the original path o f integrat ion C and the steepest descent path through the saddle points is shown for the integrals in the solu- t ion of barot ropic f low over sloping bo t tom. Figs. 8a,b,c describe the s i tuat ion for x < 2t, x = 2t and x > 2t, respectively.

155

x /2 and the signal velocity to be 2 (cf. eq. 6 and TTM, § 4.2). Having obtained all the interesting results for this case in the frame work of Laplace transforms, we now consider the path of a baroclinic current on a sloping bottom.

Baroclinic f low on a sloping bo t tom

In this section we utilize the insight gained from the simple problems discussed in the previous sections to obtain some features of the solution for a baroclinic current flowing on a sloping bot tom. In this case the solution is obtained by inversion of eqs. 45 and 46. Choosing the curvature K such that eq. 56 is satisfied, these reduce to:

~(p,t) = (U2)Up erfc(--tx/~) ~. (U2)[( 1 + p U / m i ) I i ]

2[p2(U 2) + o ~ T ] 2U~/aa + 02/4p 2 -- p2s2

(U2)(1 + p U / m 2 ) I 2

20x /o~ + 02[4p 2 -- p2s2 (A21)

o )Voo+ ml,2 --P U> + o +_ ~ --p2s2 (A22) 4p 2

11. 2 -- ~

Let:

z (_I J1,2 = ~ J x /us + a2/4p 2 --p2s2 dp {A24)

7-- ioo

The first term can be inverted exactly to yield U/2 erfc(-- tv~) cos (xx/Uo/< U2)) which again reduces to the steady-state so lu t ionaf ter t = 0 for

-+ oo. The rest of the terms describe the transient and are given in terms of J1,2 as:

<U2) ~t~an~ent = ' 2U [J1 -- Js] (A25)

Branch lines and poles We note that mi , m2 and hence J1, J~ have branch points at:

Oa + 02/4p 2 - -p2s2 = 0

i.e.:

0 ~ - + ~ ± 8 2 p 2 = 0

2s 2 (A26)

156

Imp

t - - Y -

~p

c

Fig. 9. Branch lines for invers ion integrals in the p r o b l e m of baroc l in ic f low over s loping b o t t o m .

The branch lines are chosen as indicated in Fig. 9 in which the path of integration in Jz ,2 is shown as the line C. This choice of branch lines implies that:

I Ip21< 0 2s 2

V ~ . . . . . . . . . .

= / °2 i

I m p = 0; I p2 I< o 2s 2

. . . . -2 . . . . . . . . . . . R e p <~ 0 ~ il o a + p 2 s 2 = T- p2s2 (A28) 4p 2 ~ 4p 2

One of the integrals has poles at p = +- K / a ~ ] i u2-). For the branch lines chosen, it can be shown that the poles belong to ,]1 if and only if:

2 U(U) > ( U 2) (A29)

and J2 otherwise. Again it can be shown that the sum of the residues of these poles is:

~] erfc(- - tv~) cos (x o~<_~)

Thus the solution contains the steady-state term only when the poles do not contr ibute to the integrals.

R e p = 0;

then:

I m p ~ 0 ~

157

Saddle points The saddle points of the integrals Ji.2 are determined by:

(px + mit+ m2/4{3) = 0 i = 1, 2

Using expressions (42) for ml.2 this becomes:

2~(x -- t) + ofJt/p 2 + p( + 2fJtsp 2 - 2s2p 3"] L 4p4 2P 3 2p 2 J

= -+ - r (A30) °V~-~D 2 + a~ --p2s2

This yields a 9th order equation in p for the saddle points of J1,2, viz.:

_ o 4 f J ( x - U t ) + l [oaf~2t( x _ U t ) - a4-~--4U~ pa - ~

[( xJ °'f3" a3~ 3(U2)--2U(V)) t - - 2 a + [ [ x - - ( - - 2 a ) t ] 2 + 7 2 - j - - (4~ 2 + 3 s 2 ) t 2 - - o U ( s 2 + ~)I + o~__~ [x<U~> + t(<U2> _ 2U(U>)I

0 2 + 4 ~ 2 o ( x - t ) ( ~ x - (s 2 + ~)t ) + - i [< ~ >2 _ 4(~ + s 2 )21

+ 4 p o ~ [ ~ x ( 2 - - s 2) + t(s 4 - s~ 2 - 2 ~ ( u > S ) ]

+ p~ [o~< ha> 2 _ 41~2s2 [ (x - t)" + s~t21 ]

+ 4s2{3pa[<Uu>t -- ( 2 --s2)x] - -s2<U2)2p 4 = 0 (A31)

We assume that this parameter ~ is large and classify the saddle points by using their dependence on ~, in order to facilitate the search for roots of this equation.

Saddle po in t s proport ional to {J. For large ~, the only significant terms axe of the order ~4 viz.:

--<U2 >p 4 + 4~pa [( U2>( U) t - - ((U) 2 - - 82)3:] - - 4~2p2 [(x - - ( U ) t ) 2 + s2t 2 ] = 0

The roots of this equation are:

p = 7 - + i~

2~ [t<U)(U2)__x(<U)2__S2) ] 7 - < U2)2

= 2{3s [2(U>x_<U2>t ] < U% 2

(A32)

158

These are seen to be identical to the saddle points for the flat-bottom solu- t ion (A12). If the paths of integration can be deformed so as to be the steepest descent path through their respective saddle points, then the region of instability as well as the behaviour of the solution in this region will be identical to that for baroclinic flow on a flat bot tom. Since the saddle points are at a large distance from the branch points, there should be no difficulty in transforming the path in this manner.

Saddle poin ts o f the form 1//3. These are the saddle points close to the origin. We have:

_ 0 4 / 3 ( x - U t ) + 0 3 / 3 2 t ( x - ~ t ) . . . . p5 p4 = 0 p = o/2/3t (A33)

This is seen to be identical to one of the saddle points of the barotropic meander problem (A20). Again Laplace's method yields the contribution:

° ° o ] 23/2~5/2t3 exp t 2 + ~ ( x - - ( ( U ) + ( ~ ) t ) + ~ - ~ ( (U)+a +i~b)

It is clear that the contribution due to this saddle point vanishes in the limit of large/3 except along the characteristic t = 0.

Saddle poin ts independent of/3. The equation governing the saddle points independent of/3 is

o~t(x -- Ut) + 02 [(x -- t) 2 + 4~t(x -- ( U> t) -- 3s2t 2] p4 p2

+ 4 o ( x - - t ) [ ~ x - - ( s 2 +~)t]

- - 4pas2[(x -- (U)t) 2 + s2t 2] = 0 (A34)

The six roots of this equation (which is a cubic in p2) can be obtained for a given profile U, , and for various x / t . These are shown in Fig. 10. Comparison of Figs. 9 and 10 indicates that for large x / t , four of these saddle points approach the branch points. However they never reach the branch points, since our classification of p as proportional to/3,/3 -1, fl0 fails when t becomes 0(/3-t/2).

A s y m p t o t i c s and signals The other two saddle points proportional to/30 are far from the branch

points for large x / t and are merely a modification of the barotropic saddle points of the form +ix/ot /x . For small t /x , these are located at:

_ _ t 2 p 2 ~ or+ [ o ( 4 a + U - - 2 ( U ) ) + 4 a ]

x - j

The corresponding saddle points for the barotropic case are given by

ot ot ot 2 p 2 = _ _ _ x - - t x x 2

159

2

In Ip

(

I

~t

~" I~ep

Fig. 10. Saddle points of J1.2 occurring in the solution of baroclinic flow over sloping bottom, the numbers 1, 2 near them describing whether the point belongs to J1 or J2" o, o, indicate positions at x/t = 10 and x/ t = 3 for the Gulf Stream profile in Table I, the ar- rows indicating the direction of mot ion of the saddle points with increasing t/x.

These saddle points imply that some features of the solution for small t/x will be similar to that of the barotropic case. Thus for small t, the path contains long waves whose wavelength increases with distance from the inlet. The wavelength decreases with time and meanders are formed by zeroes moving in from infinity.

The manner in which the meanders are formed for a baroclinic current resembles that of formation of barotropic meanders markedly, when (A29) is satisfied. In this case, the poles and these saddle points belong to the same integral J1. For large x/t, the poles lie between the original path and the path of steepest descent through the saddle points and the residues of the poles contribute to J1. Thus the solution does not contain the final steady-state term. As time progresses, the saddle points approach the poles and coincide with them at a certain value of x/t. From that time onwards, the solution contains a term describing the final steady state and we may define the instant at which the path of steepest descents passes through the pole to be the instant at which the signal arrives. The signal velocity is then equal to the value of x/t at which the poles coincide with the saddle points. It can be shown that the signal velocity, in fact equals the group velocity at the dominant frequency %(0) (cf. section: "Mathematical model"). Thus, when (A29) is satisfied, we expect the general nature of the solution to be modi- fied mainly by the region of instability while its asymptotics and propagation of signals etc. remain similar to the barotropic case. The modification due to the other saddle points and the branch-line integrals will be considered in the next section.

Negative group velocity We note that when (60) is not satisfied, the saddle points discussed above

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belong to J1 while the poles belong to J2. Furthermore in this case Cg(0) defined in eqs. 14 is negative. The only two saddle points belonging to J2 lie on the real p axis for all x/ t of interest. Neither the saddle points nor the path of steepest descent through them coincide with the poles for any x/t .

Contribution o f the other saddle points and branch line integrals Since the other saddle points remain close to the branch points for a large

range of t/x, it is not clear whether their contr ibut ion can be computed by using (A4). The approximation (A4) may not be too bad for the saddle points on the imaginary axis since the steepest descent paths through them make an angle of 45 ° with the axes, and thus are not along the branch lines. For these saddle points, the expression (A4) yields small amplitude waves traveling with a phase velocity which is retrograde for the Gulf Stream profile. For one of the saddle points on the real p axis, the steepest descent path is along the branch line. A more sophisticated method, would, therefore, have to be used in evaluating the contr ibution of the saddle points on the real p axis.

When the path of integration is deformed to coincide with the steepest descent paths through the various saddle points, parts of the path would be around the branch lines. Integration along such parts of the path involves evaluation of integrals of the type:

27r o

+82172 O2/4772 - - 2 sin h(tx/oa + s2772 -- o2/477 -~)

oU-- 72< U2> J

1 - ( ~ - ~< + 2s~ aV -- 17 2 (U2)

In order to analyse their contr ibution to the solution, these integrals have to be evaluated numerically, which will not be done here.

REFERENCES

Abromowitz, M. and Stegun, I.A., 1965. Handbook of Mathematical Functions. National Bureau of Standards.

Brillouin, L., 1960. Wave Propagation and Group Velocity. Academic Press, New York, N.Y.

Carrier, G.F., Krook, M. and Pearson, C.E., 1966. Functions of a Complex Variable. McGraw-Hill, New York, N.Y.

Courant, R. and Hilbert, D., 1962. Methods of Mathematical Physics, II. Interscience, New York, N.Y.

Fjortoft , R., 1951. Stabili ty properties of large-scale atmospheric disturbances. In: Com- pendium of Meteorology. American Meteorological Society, Boston, Mass., pp. 454-- 463.

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Fuglister, F.C., 1965. Gulf Stream '60. Prog. Oceanogr., 1: 265--373. Fuglister, F.C. and Worthington, L.V., 1951. Some results of a multiple ship survey of the

Gulf Stream. Tellus, 3: 1--14. Gadgil, S., 1970. Time-dependent Topographic Meandering and Jets in Rotating Systems.

Thesis, Harvard University, Cambridge, Mass. Orlanski, I., 1969. The influence of bot tom topography on stability of jets in a baroclinic

fluid. J. Atmos. Sci., 26: 1216--1232. Robinson, A.R. and Gadgil, S., 1970. Time dependent topographic meandering. Geo-

phys. Fluid Dyn., 1: 411--438. Tareev, B.A., 1965. Unstable Rossby waves and the instability of ocean currents, lzv.

Atmos. Oceanic Phys., 1(4): 426--438.

Time-dependent topographic meandering of a baroclinic current

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