advertisement

DYNAMIC OCEANOGRAPHY JAN ERIK WEBER Department of Geosciences Section for Meteorology and Oceanography University of Oslo 09.11.04 2 CONTENTS 1. SHALLOW-WATER THEORY, QUASI-HOMOGENEOUS OCEAN 1.1 Inviscid motion, potential vorticity………………………………………….3 1.2 Linear waves in the absence of rotation……………………………………..8 1.3 Effect of rotation, geostrophic adjustment…………………………………13 1.4 Sverdrup waves, inertial waves and Poincarè waves………………………16 1.5 Kelvin waves at a straight coast……………………………………………23 1.6 Amphidromic systems……………………………………...........................26 1.7 Equatorial Kelvin waves…………………………………………………...29 1.8 Topographically trapped waves……………………………………………31 1.9 Planetary Rossby waves……………………………………………………35 1.10 Topographic Rossby waves………………………………………………..41 1.11 Barotropic instability………………………………………………………42 1.12 Barotropic flow over an ocean ridge………………………………………45 2. WIND-DRIVEN CURRENTS AND OCEAN CIRCULATION 2.1 Equations for the mean motion………………………………......................51 2.2 The Ekman elementary current system……………………………………..57 2.3 The Ekman transport………………………………………………………..61 2.4 Divergent Ekman transport and forced vertical motion(Ekman suction)…..62 2.5 Variable vertical eddy viscosity…………………………………………….65 2.6 Equations for the volume transport…………………………………………68 2.7 The Sverdrup transport……………………………………………………...72 2.8 Theories of Stommel and Munk (western intensification)…………….........75 3. BAROCLINIC MOTION 3.1 Two-layer model………………………………………………………........81 3.2 Continuously stratified fluid………………………………………………..87 3.3 Free internal waves with rotation ………………………………………….88 3.4 Internal response to wind-forcing, upwelling at a straight coast…………...94 3.5 Baroclinic instability………………………………………………............100 3 1. SHALLOW-WATER THEORY, QUASI-HOMOGENEOUS OCEAN 1.1 Inviscid motion, potential vorticity Governing equations We study motion in an inviscid fluid with density. The fluid is rotating about the z-axis with constant angular velocity sin, where is the latitude of our site of observation. Furthermore, (x, y) are horizontal coordinate axes along the undisturbed sea surface, and the z-axis is directed upwards. The position of the free surface is given by z ( x, y , t ) , where is referred to as the surface elevation. The atmospheric pressure at the surface is denoted by PS(x, y, t). The bottom topography does not vary with time, and is given by z H ( x, y ) ; see the sketch in Fig. 1.1. Fig 1.1. Definition sketch. The equation of motion for a fluid particle of unit mass can now be written Dv 1 fk v p gk , (1.1.1) dt where D/dt /t + v is the total derivative following a fluid particle, and f = 2sin is the Coriolis parameter. We take that the mass of fluid within a geometrically fixed volume only can change as a result of advection of fluid particles. The basic assumption of conservation of mass for a fluid can be stated mathematically as 4 D v . dt (1.1.2) This is often referred to as the continuity equation. In this section we will assume that the density of a particle is conserved (D/dt = 0). Thus, the continuity equation reduces to v 0. (1.1.3) Furthermore we will assume that the density is the same for all particles (homogeneous fluid). We orientate our coordinate system such that the x-axis is tangential to a latitudinal circle and the y-axis is pointing northwards; see Fig. 1.2. Fig. 1.2. Orientation of coordinate axes. In our reference system f is only a function of y. We may then write approximately that df f f 0 y f 0 y , dy 0 (1.1.4) where f 0 2 sin 0 , 1 d 2 sin 0 R d 2 cos 0 . R (1.1.5) This is called the beta-plane approximation. If f is approximately constant in an (x, y)area, we say that the motion occurs on an f-plane. The kinematic and dynamic boundary conditions at the surface can be written, respectively, as 5 D ( z ) 0, z ( x, y , t ) , dt (1.1.6) p PS , z ( x, y , t ) . (1.1.7) The kinematic boundary condition at the bottom can be expressed as v z H 0, z H ( x, y ) . (1.1.8) We now integrate the continuity equation (1.1.3) from z = H to z = (x, y, t): wz dz u x dz v y dz . H H (1.1.9) H Utilizing (1.1.6) and (1.1.8), we find that t u dz v dz . x H y H (1.1.10) The basic assumption in shallow water theory is that the pressure distribution in the vertical direction is hydrostatic. By utilizing that the density is constant, and applying (1.1.7), this leads to p g ( z ) PS ( x, y, t ) . (1.1.11) This means, when we return to the vertical component in (1.1.1), that the vertical acceleration Dw/dt must be so small that it does not noticeably alter the hydrostatic pressure distribution. We will return to the validity of this in section 1.2. The horizontal components of (1.1.1) can thus be written Du 1 fv g x PSx dt (1.1.12) Dv 1 fu g y PSy dt (1.1.13) Throughout this text we will alternate between writing partial derivatives in full, and (for economic reasons) as subscripts. We realize that the right-hand sides of (1.1.12) and (1.1.13) are independent of z. By utilizing that v Dy/dt and f = f0 + y, (1.1.12) can be written D 1 1 2 u f 0 y y g x PSx dt 2 (1.1.14) From (1.1.14) it follows that D(u f 0 y y 2 / 2) / dt is independent of z. Thus, this is also true for (u f 0 y y 2 / 2) , and thereby also for u, if u and v were independent of 6 z at time t = 0. Similarly, from (1.1.13) we find that v is independent of z. We can accordingly write u u ( x, y, t ), v v( x, y, t ). (1.1.15) Furthermore, it now follows from (1.1.3) that wz is independent of z. Hence, by integrating in the vertical: w (u x v y ) z C ( x, y, t ) . (1.1.16) The function C is obtained by applying the boundary condition (1.1.8) at the ocean bottom. The vertical velocity can thus be written w (u x v y )( z H ) uH x vH y . (1.1.17) Since u and v are independent of z, the integrations in (1.1.10) are easily performed. We then obtain the following nonlinear, coupled set of equations for the horizontal velocity components and the surface elevation ut uu x vuy fv g x 1 PSx , (1.1.18) PSy , (1.1.19) t u( H )x v ( H )y 0 . (1.1.20) vt uv x vv y fu g y 1 To solve this set of equations we require three initial conditions, e.g. the distribution of u, v, and in space at time t = 0. If the fluid is limited by lateral boundaries (walls), we must in addition ensure that the solutions satisfy the requirements of no flow through impermeable walls. Potential vorticity We define the vertical component of the relative vorticity in our coordinate system, e.g. Fig. 1.2, by vx u y . (1.1.21) In addition, every particle in this coordinate system possesses a planetary vorticity f, arising from solid body rotation with angular velocity sin . Hence, the absolute vertical vorticity for a particle becomes f . We shall derive an equation for the absolute vorticity. It is obtained by differentiating the equations (1.1.18) and (1.1.19) by / y and / x , respectively, and then add the resulting equations. 7 Mathematically, this means to operate the curl on the vector equation to eliminate the gradient terms. Since f is independent of time, we find that D ( f ) ( f )( u x v y ) . dt (1.1.22) By using that H is independent of time (1.1.20) can be written D ( H ) ( H )( u x v y ) . dt (1.1.23) Here, H + is the height of a vertical fluid column. We define the potential vorticity Q by Q f . H (1.1.24) By eliminating the horizontal divergence between (1.1.22) and (1.1.23), we find for Q that DQ 0. dt (1.1.25) This equation expresses the fact that a given material vertical fluid column always moves in such a way that its potential vorticity is conserved. Alternatively, we can apply Kelvin’s circulation theorem for an inviscid fluid to derive this important result. Kelvin’s theorem states that the circulation of the absolute velocity around a closed material curve (always consisting of the same fluid particles) is conserved. For a material curve in the horizontal plane, Kelvin’s and Stokes’ theorems yield v r k abs ( vabs ) const . , (1.1.26) where is the area inside . Furthermore, in the surface integral: k ( vabs ) f . (1.1.27) When the surface area in (1.1.26) approaches zero, we have ( f ) const. (1.1.28) In addition, the mass of a vertical fluid column with base must be conserved, and hence ( H ) const. (1.1.29) This is valid for all times, since a vertical fluid column will remain vertical; see (1.1.15). In our case the fluid is homogeneous and incompressible, i.e. is the same 8 for all particles. Thus, by eliminating between (1.1.28) and (1.1.29), we find as before that Q f const. , H (1.1.30) or, equivalently, DQ / dt 0 . In the ocean we usually have that || << f and || << H. For stationary flow, assuming that |H| >> || and |f H| >> |H |, (1.1.25) yields approximately that v ( f / H ) 0 . (1.1.31) On an f-plane, this equation reduces to v H 0 . (1.1.32) Accordingly, the flow in this case follows the lines of constant H (i.e. the bottom contours). This phenomenon is called topographic steering. On a beta-plane the flow will follow the contours of the function f/H; see (1.1.31). Take that the motion is approximately geostrophic. For constant surface pressure, we then obtain from (1.1.18) and (1.1.19) that g v (k ) . f (1.1.33) For stationary flow the theorem of conservation of potential vorticity reduces to v Q 0 . Combined with (1.1.33), this leads to k ( Q ) 0 . (1.1.34) This means that the surfaces of constant and constant Q coincide. Accordingly, Q and are uniquely related, or (Q ) . (1.1.35) For the special case where we have topographic steering on an f-plane (when Q varies only with H), we find from (1.1.35) that (H ) . (1.1.36) Accordingly, for stationary, geostrophic motion with a free surface, the iso-lines for the surface elevation are parallel to the depth contours. 1.2 Linear waves in the absence of rotation We assume small disturbances from the state of equilibrium in the ocean, twodimensional motion (/y = 0, v = 0), and constant depth. Furthermore, we take the 9 atmospheric pressure to be constant along the surface. The set of equations (1.1.18)(1.1.20) can then be linearized, reducing to ut g x , t Hu x . (1.2.1) Eliminating the horizontal velocity, we find tt gH xx 0 . (1.2.2) This equation is called the wave equation. By inserting a wave solution of the type exp( ik ( x c0t ) , representing a complex Fourier component, we find from (1.2.2) that c0 (gH )1 / 2 , i.e. all wave components move with the same phase speed regardless of wavelength. Such waves are called non-dispersive. We need not limit ourselves to consider one single Fourier component. From (1.2.2) we realize immediately that a general solution can be written F1 ( x c0t ) F2 ( x c0t ) . (1.2.3) If, at time t = 0, the surface elevation was such that = F(x), and t = 0, it is easy to see that the solution becomes 1 F ( x c0t ) F ( x c0t ). 2 (1.2.4) From (1.2.1) and (1.2.4) we find for the acceleration ut g x g F ' ( x c0t ) F ' ( x c0t ), 2 (1.2.5) where F ' ( ) dF / d . Hence, the horizontal velocity is given by u g F ( x c0t ) F ( x c0t ). 2c0 (1.2.6) From (1.2.4) we can display the evolution of an initially bell-shaped surface elevation F(x) with typical width L; see the sketch in Fig. 1.3. 10 Fig. 1.3. Evolution of a bell-shaped surface elevation. We note that the initial elevation splits into two identical pulses moving right and left with velocity c0 = (gH)1/2. In a deep ocean (H = 4000 m), the phase speed is c0 200 m s1, while in a shallow ocean (H = 100 m) we have c0 30 m s1. If the maximum initial elevation in this example is h, i.e. F(0) = h, we find from (1.2.6) that the velocity in the ocean directly below peak of the right-hand pulse can be written u gh , 2c 0 (1.2.7) when t >> L/c0, that is after the two pulses have split. If we take h = 1 m as a typical value, the deep ocean example yields u 2.5 cm s1, while for the shallow ocean we find u 17 cm s1. As a second example we consider an initial step function: 1 2 h, x 0, F ( x) 1 h, x 0. 2 (1.2.8) In this case, the velocity and amplitude development becomes as sketched in Fig. 1.4. 11 Fig. 1.4. Evolution of a surface step function. It is obvious that we in an example like this (with a step in the surface at t = 0) must be careful when using linear theory, which requires small gradients. In a more realistic example where differences in height occurs, the initial elevation will have a final (an quite small) gradient around x = 0. Qualitatively, however, the solution becomes as discussed above. It is easy to show that the solution for the step problem satisfies the mass and energy conservation laws: Choose a geometrically fixed area –D x D, where D c0t. From Fig. 1.4 it is obvious that the mass of fluid within this area is constant. We choose z = h/2 as the level where the potential energy is zero. The initial potential energy can thus be written E p0 MghG 1 gDh 2 , 2 (1.2.9) where M = Dh is the fluid mass above the zero level, and hG = h/2 is the vertical distance to the centre of mass of this fluid. The initial kinetic energy is zero, since we start the problem from rest. At a later time t = D/c0, the potential energy becomes E p Mg while the kinetic energy can be written h 1 gDh 2 , 4 4 (1.2.10) 12 Ek 1 1 (2 DH )u 2 gDh 2 . 2 4 (1.2.11) Thus, from (1.2.9)-(1.2.11) we find that Ek E p E p0 , i.e. the total energy is conserved. This is of course also true for the bell-shaped elevation in the first example. The hydrostatic approximation Finally, we address the validity of the hydrostatic approximation in the case of waves in a non-rotating ocean. We rewrite the pressure as a hydrostatic part plus a deviation: p g ( z ) PS p' , (1.2.12) where p ' is the non-hydrostatic deviation. The vertical component of (1.1.1) becomes to lowest order: wt 1 pz , (1.2.13) while the horizontal component can be written ut g x 1 px . (1.2.14) The hydrostatic assumption implies that 1 px ut . (1.2.15) If the typical length scales in the x- and z-directions are L and H, respectively, we obtain from the continuity equation that u ~ L w, H (1.2.16) where ~ means order of magnitude. From (1.2.13) we then find p' ~ H2 ut . L (1.2.17) Utilizing this result, the condition (1.2.15) reduces to H 2 / L2 1 . (1.2.18) Thus, we realize that the assumption of a hydrostatic pressure distribution in the vertical requires that the horizontal scale L of the disturbance must be much larger 13 than the ocean depth. For a wave, L is associated with the wavelength; for a single pulse, L corresponds to the characteristic pulse width. 1.3 Effect of rotation, geostrophic adjustment We now consider the effect of the earth’s rotation upon wave motion. Linear theory still applies, and we take the depth and the surface pressure to be constant. Furthermore, we assume that f is constant. Equations (1.1.18)-(1.1.20) then reduces to ut fv g x , (1.3.1) vt fu g y , (1.3.2) t H (u x v y ) 0 . (1.3.3) We compute the vertical vorticity and the horizontal divergence, respectively, from (1.3.1) and (1.3.2). By utilizing (1.1.3), we then obtain (v x u y ) t f t 0 , H (1.3.4) and tt H f ( v x u y ) g ( xx yy ) . (1.3.5) The vorticity equation can be integrated in time, i.e. vx u y f f v 0 x u0 y 0 , H H (1.3.6) where sub-zeroes denote initial values. We assume that the problem is started from rest, which means that there are no velocities or velocity gradients at t = 0. Thus vx u y f (0 ) . H (1.3.7) Inserting for the vorticity in (1.3.5), we find that tt c02 (xx yy ) f 2 f 20 , (1.3.8) where c02 gH , and 0 is a known function of x and y (the surface elevation at t = 0). The solution to (1.3.8) can be written as a sum of a transient (free) part and a stationary (forced) part ~( x, y, t ) ( x, y ) , (1.3.9) where ~ and fulfil, respectively ~tt c02 (~xx ~yy ) f 2~ 0 , (1.3.10) 14 c02 ( xx yy ) f 2 f 20 . (1.3.11) Equation (1.3.10) for the transient, free solution is called the Klein-Gordon equation and occurs in many branches in physics. Here, it describes long surface waves that are modified by the earth’s rotation (Sverdrup waves). These waves will be discussed in the next section. Notice that the initial conditions for the free solution are ~( x, y,0) ( x, y ) ( x, y ) and ~ 0 . 0 t As an example of a stationary solution of (1.3.8), we return to the problem in the last section where the surface elevation initially was a step function: h 2 , x 0, h 2 , x 0, 0 ( x ) (1.3.12) or, for simplicity, 1, x 0, 1, x 0. 1 2 0 ( x) h sgn( x), sgn( x) (1.3.13) We assume that the motion is independent of the y-coordinate. From (1.3.11) we then obtain 1 2 xx a 2 a 2 h sgn( x) , (1.3.14) where a c0 / f . It is easy to show that the solution of (1.3.14) is given by 1 2 h{1 exp( x / a)}sgn( x) . We have sketched this solution in Fig. 1.5. Fig. 1.5 Geostrophic adjustment of a free surface. (1.3.15) 15 The distance a = c0/f = (gH)1/2/f is called the Rossby radius of deformation, or the Rossby radius for short. A typical value for f at mid latitudes is 104s1. For a deep ocean (H = 4000 m), we find that a 2000 km, while for a shallow ocean (H = 100 m), a 300 km. From (1.3.1) and (1.3.2) we find the velocity distribution for this example, i.e. f v g x , (1.3.16) u 0. (1.3.17) and We note from (1.3.16) that we have a balance between the Coriolis force and the pressure-gradient force (geostrophic balance) in the x-direction. From (1.3.15) and (1.3.16) the corresponding geostrophic velocity in the y-direction can be written v gh exp( x / a ) . 2c0 (1.3.18) This is a “jet”-like stationary flow in the positive y-direction. Although the geostrophic adjustment occurs within the Rossby radius, we notice from (1.3.18) that the maximum velocity in this case is independent of the earth’s rotation. By comparison with (1.2.7), we see that our maximum velocity it is the same as the velocity below a moving pulse with height h/2, or as the velocity in the non-rotating step-problem. Let us compute the kinetic and the potential energy within a geometrically fixed area D x D for the stationary solution (1.3.15)-(1.3.18), which is valid when t . The kinetic energy becomes D 2 1 1 Ek v dz dx gh 2 a (1 e 2 D / a ) , 2 D H 8 (1.3.19) where we have used the fact that H . For the potential energy we find h / 2 1 1 E p g z ' dz ' dx gh 2 D gh 2 a(3 4e D / a e 2 D / a ) , 2 8 D 0 D (1.3.20) where we have taken z = h/2 as the level of zero potential energy, and introduced z' z h / 2 . Initially, the total mechanical energy within the considered area equals the potential energy, or E0 E p0 1 gh 2 D . 2 Let us choose D >> a. We then notice from (1.3.19)-(1.3.21) that (1.3.21) 16 Ek E p E0 . (1.3.22) Thus, when t , the total mechanical energy inside the considered area is less than it was at t = 0. The reason is that energy in the form of free Sverdrup waves (solutions of the Klein-Gordon equation) has “leaked” out of the area during the adjustment towards a geostrophically balanced steady state. We will consider these waves in more detail in the next section. Finally we discuss in a quantitative way when it is possible to neglect the effect of earth’s rotation on the motion. For this to be possible, we must have that (1.3.23) vt fk v . Accordingly, the typical timescale T for the motion must satisfy T 2 . f (1.3.24) At mid latitudes we typically have 2/f 17 hours. If the characteristic horizontal scale of the motion is L and the phase speed is c ~ (gH)1/2, we find from (1.3.24) that the effect of earth’s rotation can be neglected if L a , (1.3.25) where a = c/f is the Rossby radius of the problem. In the open ocean L will be associated with the wavelength, while in a fjord or canal, L will be the width. Oppositely, when L a, (1.3.26) the effect of the earth’s rotation on the fluid motion can not be neglected. 1.4 Sverdrup waves, inertial waves and Poincaré waves We consider long surface waves in a rotating ocean of unlimited horizontal extent. Such waves are often called Sverdrup waves (Sverdrup, 1927). They are solutions of the Klein-Gordon equation (1.3.10). Actually, Sverdrup’s name is usually related to friction-modified, long gravity waves, but here we will use it also for the frictionless case. In literature long waves in an inviscid ocean are often called Poincaré waves. However, this term will be reserved for a particular combination of Sverdrup waves that can occur in canals with parallel walls. Sverdrup waves A surface wave component in a horizontally unlimited ocean can be written 17 A exp( i (kx ly t )) . (1.4.1) This wave component is a solution of (1.3.10) if 2 f 2 c02 (k 2 l 2 ) . (1.4.2) Here k and l are real wave numbers in the x- and y-direction, respectively, and is the wave frequency. Equation (1.4.2) is the dispersion relation for inviscid Sverdrup waves. From this relation we note that the Sverdrup wave must always have a frequency that is larger than (or equal to) the inertial frequency f. For simplicity we let the wave propagate along the x-axis, i.e. l = 0. The phase speed now becomes 2 c c0 1 2 2 k 4 a 1/ 2 , (1.4.3) where = 2/k is the wavelength and a = c0/f is the Rossby radius. We note that the waves become dispersive due to the earth’s rotation, i.e. the phase speed depends on the wavelength (here: increases with increasing wave length). The group velocity becomes cg c0 d . 1/ 2 dk 2 1 2 2 4 a (1.4.4) We notice that the group velocity decreases with increasing wavelength. From (1.4.3) and (1.4.4) we realize that ccg = c02, i.e. the product of the phase and group velocities are constant. From (1.4.2), with l = 0, we can sketch the dispersion diagram for positive wave numbers; see Fig. 1.6. Fig. 1.6. The dispersion diagram for Sverdrup waves. 18 For k << a1 (i.e. >> a) we have that f. This means that the motion is reduced to inertial oscillations in the horizontal plane. For k >> a1 gravity dominates, i.e. c0k, and we have surface gravity waves that are not influenced by the earth’s rotation. Contrary to gravity waves in a non-rotating ocean, the Sverdrup waves discussed here do possess vertical vorticity. For a wave solution ( exp( i t ) ), (1.3.4) yields f . H (1.4.5) If we still assume that /y = 0, we obtain from (1.4.5) and (1.3.2) that f , H 1 u vt . f vx (1.4.6) With in (1.4.1) depending only on x and t, we find, when we let the real part represent the physical solution: 0 cos( kx t ), u 0 cos( kx t ), kH 0 f v sin( kx t ), kH zH w 0 sin( kx t ). H (1.4.7) Here the vertical velocity w has been obtained from (1.1.17). Since f for Sverdrup waves, we must have that u v . Furthermore, from (1.4.7) we find that u2 v2 1. 0 /( kH ) 2 0 f /( kH ) 2 (1.4.8) This means that the horizontal velocity vector describes an ellipsis where the ratio of the major axis to the minor axis is / f . From (1.4.7) it is easy to see that the velocity vector turns clockwise, and that one cycle is completed in time 2/. Sverdrup (1927) demonstrated that the tidal waves on the Siberian continental shelf were of the same type as the waves discussed here. In addition, they were modified by the effect of bottom friction, which leads to a damping of the wave amplitude as the wave progresses. Furthermore, friction acts to reduce of the phase speed, and it causes a phase displacement between maximum current and maximum surface elevation. 19 Inertial waves When discussing the dispersion relation (1.4.2), we realized that for very long wavelengths the Sverdrup wave was reduced to an inertial wave. For the inertial wave phenomenon, the effect of gravity is unimportant. Hence, to study such waves we may take that the surface is horizontal at all times, i.e. 0. Furthermore, for infinitely long waves we must put / x 0, / y 0 in the governing equations (1.3.1)(1.3.3). They then reduce to ut fv 0, vt fu 0. (1.4.9) d2 2 f 2 (u, v) 0 , dt (1.4.10) Accordingly, where we have changed to ordinary derivatives since u and v now are only functions of time. With initial conditions u = u0, v = 0, the solution becomes u u0 cos ft, v u0 sin ft. (1.4.11) We notice from (1.4.11) that u 2 v 2 u 02 , i.e. the magnitude of the velocity is u0 everywhere in the ocean, and that the velocity vector turns clockwise in the northern hemisphere (where f > 0). The trajectory of a single fluid particle (xp, yp) can be found from the relations u = dxp /dt and v = dyp /dt, where u and v are given by (1.4.11). Accordingly u0 sin ft, f u y p y0 0 cos ft, f x p x0 (1.4.12) u02 . f2 (1.4.13) or ( x p x0 )2 ( y p y0 )2 This means that a fluid particle moves in a closed circle around the point (x0, y0), which is the mean position of the particle in question. The radius of the circle is r = u0/f, which is called the inertial radius. If we use the typical values u0 = 10 cm s1 and f = 104s1, we find that r = 1 km. The inertial period T = 2/f, which is the time 20 needed for a particle to make one closed loop, is about 17 hours in this example. The particle moves in a clockwise direction in the northern hemisphere. Since the vertical component of the earth’s rotation at a location with latitude is sin, we can introduce the pendulum day T defined by T 2 . sin (1.4.14) We notice that the inertial period is half a pendulum day. It turns out that inertial oscillations are quite common in the ocean. Most ocean current spectra show a local amplitude or energy maximum at the inertial frequency. Poincaré waves We consider waves in a uniform canal along the x-axis with depth H and width B. Such waves must satisfy the Klein-Gordon equation (1.3.10). But now the ocean is laterally limited. At the canal walls, the normal velocity must vanish, i.e. v = 0 for y = 0, B. By inspecting (1.4.7), we realize that no single Sverdrup wave can satisfy these conditions. However, if we superimpose two Sverdrup waves, both propagating at oblique angles ( and , say) with respect to the x-axis, we can construct a wave which satisfies the required boundary conditions. The velocity component in the ydirection must then be of the form v v0 sin( n y ) exp( i (kx t )), n 1,2, 3,.. B (1.4.15) Since the wave number l n / B in the y-direction now is discrete due to the boundary conditions, the dispersion relation (1.4.2) becomes 1/ 2 2 n 2 2 2 2 f c0 k 2 , n 1,2,3, ... B (1.4.16) We notice from (1.4.15) that the spatial variation in the cross-channel direction is trigonometric. Such trigonometric waves in a rotating channel are called Poincaré waves. They can propagate in the positive as well as the negative x-direction. We shall see that this is in contrast to coastal Kelvin waves, which we discuss later in section 1. In general, the derivation of the complete solution for Poincaré waves is too lengthy to be discussed in this text. For a detailed derivation; see for example LeBlond and Mysak (1978), p. 270. 21 Energy considerations By utilizing the solution (1.4.7), we can compute the mechanical energy associated with Sverdrup waves. The mean potential energy per unit area of a fluid column can be written 1 Ep 2 / 2 / 1 ( g z dz)dt 4 g 0 2 0 . (1.4.17) 0 The mean kinetic energy per unit area becomes Ek 1 2 / 2 / 0 1 1 1 f 2 / 2 2 u 2 v 2 w2 dz}dt g 0 , 2 4 1 f 2 / 2 H 0 { (1.4.18) where we have utilized that kH << 1. We see that in a rotating ocean (f 0), the mean potential and the mean kinetic energy in the wave motion are no longer equal (compare with the results (1.2.10) and (1.2.11) for the non-rotating case, where we have an equal partition between the two). The dominating part of the mean energy is now kinetic. The total mean mechanical energy becomes E Ek E p 1 g02c 2 / c02 . 2 (1.4.19) Energy flux and group velocity The formula for the group velocity, c g d / dk , used in (1.4.4), arises from purely kinematic considerations. By superimposing two progressive wave trains with the same amplitude and with wave numbers k and k +k, and frequencies and +, respectively, one finds that the envelope containing the wave crests and wave troughs, defined as the wave group, will propagate with the speed / k . However, the group velocity has also a very important dynamical interpretation. Consider a Sverdrup wave that propagates along x-axis. This wave induces a net transport of energy in the x-direction. The energy transport through a vertical cross-section can be found by computing the mean work done by the fluctuating pressure at that section. The mean work F , per unit time and unit area, which is performed on the fluid to the right of a vertical cross-section, can be written 1 F 2 / 2 / 0 H ( pu dz)dt , (1.4.20) where p is the fluctuating part of the pressure in the fluid. Inserting from (1.1.11) and (1.4.7), it follows that 22 F 1 g02c . 2 (1.4.21) For a regular, infinitely long wave train the mean energy cannot accumulate in space. Hence there must be a transport of mean total energy E through the considered cross-section. Denote this transport velocity by ce. Energy balance then requires that F ce E . (1.4.22) Equation (1.4.22) yields, by inserting from (1.4.19) and (1.4.21), that ce c02 / c c g , (1.4.23) where the last equality follows from (1.4.4). Accordingly, the mean energy in the wave motion propagates with the group velocity. The work done by the fluctuating pressure per unit mass and unit time is often referred to as the energy flux, and the total energy per unit mass as the energy density. These quantities vary in space and time. Both concepts follow naturally from the energy equation for the fluid. With no variation in the y-direction, the linearized equations (1.3.1)-(1.3.3) reduce to ut fv g x , vt fu 0, t Hu x . (1.4.24) By multiplying the two first equations by u and v, respectively, and then adding, we obtain 1 2 (u v 2 ) ( gu ) gu x . t 2 x (1.4.25) Obviously, the Coriolis force does not perform any work since it acts perpendicular to the displacement (or velocity). By inserting from the continuity equation that u x t / H into the last term, (1.4.25) becomes 1 2 g 2 2 u v ( gu ) 0 . t 2 H x (1.4.26) We write this equation ed e f 0 , t x (1.4.27) where the energy density ed and the energy flux ef are defined, respectively, as 1 g ed ( u 2 v 2 2 ) , 2 H (1.4.28) 23 e f gu . (1.4.29) The mean values for a vertical fluid column become 2 / 1 1 02 2 ed ( e dz ) dt g , d 2 / 0 H 2 k 2c02 2 / 1 1 2 ef ( e f dz )dt g0 / k . 2 / 0 H 2 (1.4.30) By comparing with (1.4.19) and (1.4.21) we realize that ed E / and e f F / , and hence e f c g ed . Even though we have only shown this to be valid for Sverdrup waves, it is a quite general result, and valid for all kinds of wave motion. 1.5 Kelvin waves at a straight coast We consider an ocean that is limited by a straight coast. The coast is situated at y = 0; see Fig. 1.7. Fig. 1.7. Definition sketch. Furthermore, we assume that the velocity component in the y-direction is zero everywhere, i.e. v 0. With constant depth and constant surface pressure (1.4.24) becomes ut g x , (1.5.1) fu g y , (1.5.2) t Hu x . (1.5.3) 24 We take that the Coriolis parameter is constant, and eliminate u from the problem. Equations (1.5.1) and (1.5.2) yield yt f x 0 , (1.5.4) t ac0 xy 0 . (1.5.5) while (1.5.2) and (1.5.3) yield Here c0 (gH )1 / 2 and a = c0/f. We assume a solution of the form G ( y ) F ( x, t ) . (1.5.6) Ft aG' , c0 Fx G (1.5.7) By inserting into (1.5.5), we find where G ' dG / dy . The left-hand side of (1.5.7) is only a function of x and t, and the right-hand side is only a function of y. Thus, for (1.5.7) to be valid for arbitrary values of x, y, and t, both sides must equal to the same constant, which we denote by ( 0 for a non-trivial solution). Hence aG' G Ft c0 Fx G exp( y / a), F F ( x c0t ). (1.5.8) By inserting from (1.5.8) into (1.5.4), we find that 1 . (1.5.9) Accordingly, from (1.5.8), we have solutions of the form exp( y / a) F ( x c0t ) , (1.5.10) exp( y / a) F ( x c0t ) . (1.5.11) and If we have a straight coast at y = 0 and an unlimited ocean for y > 0, as depicted in Fig. 1.7, the solution (1.5.10) must be discarded. This is because must be finite everywhere in the ocean, even when y . The solution for the surface elevation and the velocity distribution in this case then become exp( y / a ) F ( x c0t ), u g exp( y / a ) F ( x c0t ). fa (1.5.12) This type of wave is called a single Kelvin wave (double Kelvin waves will be treated in section 1.8). It is trapped at the coast within a region determined by the Rossby 25 radius. It is therefore also referred to as a coastal Kelvin wave. The Kelvin wave propagates in the positive x-direction with velocity c0, like a gravity wave without rotation. The difference from the non-rotating case, however, is that now we do not have the possibility of a wave in the negative x-direction. This is because the Kelvin wave solution requires geostrophic balance in the direction normal to the coast; see (1.5.2). This is impossible for a wave in the negative x-direction in the northern hemisphere. In general, if we look in the direction of wave propagation (along the wave number vector), a Kelvin wave in the northern hemisphere always moves with the coast to the right, while in the southern hemisphere (f < 0), it moves with the coast to the left; see the sketch in Fig. 1.8 for a single Fourier component in the northern hemisphere. Fig. 1.8. Propagation of Kelvin waves along a straight coast when f > 0. Since the wave amplitude is trapped within a region limited by the Rossby radius, the wave energy is also trapped in this region. The energy propagation velocity (the group velocity) is here c g d / dk d (c0 k ) / dk c0 , and the energy is propagating with the coast to the right in the northern hemisphere. We note that for Kelvin waves the frequency has not a lower limit (for Sverdrup waves f). The oceanic tide may in certain places manifest itself as coastal Kelvin waves of the type studied here. We will discuss this further in connection with amphidromic points (points where the tidal height is always zero). From (1.5.12) we notice that the surface elevation and velocity are in phase, i.e. maximum high tide coincides with maximum current. It turns out from measurements that the maximum tidal current at a given location occurs before maximum tidal height. This is due to the effect of friction at the ocean bottom, which we have neglected so far. 26 The effect of bottom friction will also modify the Kelvin wave solution (1.5.12) in various other ways. In particular, it turns out that the lines describing a constant phase (the co-tidal lines) are no longer directed perpendicular to the coast, but are slanting backwards relative to the direction of wave propagation (Martinsen and Weber, 1981). This situation is sketched in Fig. 1.9. Fig. 1.9. Coastal Kelvin waves influenced by friction. Other frictional effects on Kelvin waves are reduced phase speed (c < c0), and an amplitude decrease as the wave progresses, which is similar to how friction affects Sverdrup waves. 1.6 Amphidromic systems Wave systems, where the lines of constant phase, or the co-tidal lines, form a star-shaped pattern, are called amphidromies. They are wave interference phenomena, and in the ocean they usually originate due to interference between Kelvin waves. Let us study wave motion in an ocean with width B; see the sketch in Fig. 1.10. Fig. 1.10. Ocean with parallel boundaries (infinitely long canal). 27 Since the ocean now is limited in the y-direction, both Kelvin wave solutions (1.5.10) and (1.5.11) can be realized. Because we are working with linear theory, the sum of two solutions is also a solution, i.e. exp( y / a) F ( x c0t ) exp( y / a) F ( x c0t ) . (1.6.1) In general the F-functions in (1.6.1) can be written as sums of Fourier components. It suffices here to consider two Fourier components with equal amplitudes: Aexp( y / a) sin( kx t ) exp( y / a) sin( kx t ) , (1.6.2) where = c0k. Along the x-axis, i.e. for y = 0, (1.6.2) reduces to 2 A sin kx cos t . (1.6.3) This constitutes a standing oscillation with period T = 2/. Zero elevation ( = 0) occur when x n , n 0,1, 2, ... k (1.6.4) At the locations given by (n/k, 0), the surface elevation is zero at all times. These nodal points are referred to as amphidromic points. We consider the shape of the co-phase lines, and choose a particular phase, e.g. a wave crest (or trough). At a given time the spatial distribution of this phase is given by t 0 ; i.e. a local extreme for the surface elevation. Partial differentiation (1.6.2) with respect to time yields that the co-phase lines are given by the equation exp( y / a) cos( kx t ) exp( y / a) cos( kx t ) 0 . (1.6.5) We notice right away that the co-phase lines must intersect at the amphidromic points x n / k , y 0 for all times. As an example, we consider the amphidromic point at the origin. In a sufficiently small distance from origin, x and y are so small that we can make the approximations exp( y / a ) 1 y / a, cos kx 1, sin kx kx . Equation (1.6.5) then yields y (ka tan t ) x . (1.6.6) This means that the co-phase lines are straight lines in a region sufficiently close to the amphidromic points. Since tan t is a monotonically increasing function of time in the interval t 0 to t /( 2 ) , we see that a co-phase line revolves around the amphidromic point in a counter-clockwise direction in this example (f > 0). It turns out that, as a main rule, the co-phase lines of the amphidromic systems in the world oceans rotate counter-clockwise in the northern hemisphere and clockwise in the 28 southern hemisphere. We notice from (1.6.6) that if we have high tide along a line in the region x > 0, y > 0 at some time t, we will have high tide along the same line in the region x < 0, y < 0 at time t / , or half a period later. We now consider the numerical value of along a co-phase line. Close to an amphidromic point, (here the origin), we can use (1.6.2) to express the elevation as y a 2 A(kx cos t sin t ) . (1.6.7) From (1.6.6) we find that tan t y /( kax) along a co-phase line. By eliminating the time dependence between this expression and (1.6.7), we find for the magnitude of the surface elevation along a co-tidal line: 2 Ak 2 x 2 y 2 / a 2 . 1/ 2 (1.6.8) The lines for a given difference between high and low tide are called co-range lines. These curves are given by (1.6.8), when is put equal to a constant, i.e. k 2 x 2 y 2 / a 2 const . (1.6.9) We thus see that the co-range lines close to the amphidromic points are ellipses. In Fig. 1.11 we have depicted co-phase lines (solid curves) and co-range lines (broken curves) resulting from the superposition of two oppositely travelling Kelvin waves, both with periods of 12 hours and amplitudes of 0.5 m. The wavelength is 800 km, the width of the channel is 400 km, the depth is 40 m, and the Coriolis parameter is 104s1. The Rossby radius becomes 198 km in this example. Hence the right-hand side of the channel is dominated by the upward-propagating Kelvin wave (the one with minus sign in the phase), and the left-hand side is dominated by the downward propagating Kelvin wave. 29 Fig. 1.11. Amphidromic system in an infinitely long canal. 1.7 Equatorial Kelvin waves Close to equator we have that f0 0. The Coriolis parameter in this region can then be approximated by f y , (1.7.1) where the y-axis is directed northwards; see Fig. 1.12. Fig. 1.12. Sketch of the co-ordinate axes near the equator. We will find that it is possible to have equatorially trapped gravity waves, analogous to the trapping at a straight coastline. Assume that the velocity component in the ydirection is zero everywhere, i.e. we assume geostrophic balance in the direction 30 perpendicular to equator. With constant depth, the equations (1.5.1)-(1.5.3) are unchanged, but now f y in (1.5.2). By assuming a solution of the form G ( y ) F ( x, t ) as before, (1.5.7) becomes Ft c G' 0 . c0 Fx yG (1.7.2) Accordingly: F F ( x c0t ), 2 G exp y . 2c0 (1.7.3) By inserting into (1.5.4), we find 1 . (1.7.4) From (1.7.3) we realize that to have finite solution when y , we must choose = 1 in (1.7.4). The solution thus becomes g 2 2 u exp( y / ae ) F ( x c0t ), c0 exp( y 2 / ae2 ) F ( x c0t ), (1.7.5) where the equatorial Rossby radius is defined by ae (2c0 / )1 / 2 . (1.7.6) We note that the solution (1.7.5), referred to as an equatorial Kelvin wave, is valid at both sides of equator and that it propagates in the positive x-direction, i.e. eastwards with phase speed c0 = (gH)1/2. The energy also propagates eastwards with the same velocity, since we have no dispersion. At the equator is approximately 21011m1s1. For a deep ocean with H = 4000 m, we find from (1.7.6) that the equatorial Rossby radius becomes about 4500 km. Equatorial Kelvin waves are generated by tidal forces, and by wind stress and pressure distributions associated with storm events with horizontal scales of thousands of kilometres. When such waves meet the eastern boundaries in the ocean (the west coast of the continents), part of the energy in the wave motion will split into a northward propagating coastal Kelvin waves in the northern hemisphere, and a southward propagating coastal Kelvin wave in the southern hemisphere. Some of the energy will also be reflected in the form of planetary Rossby waves. This type of wave will be discussed in section 1.9. 31 1.8 Topographically trapped waves We have seen that gravity waves can be trapped at the coast or at the equator due to the effect of the earth’s rotation. Trapping of wave energy in a rotating ocean can also occur in places where we have changes in the bottom topography. In this case, however, the wave motion is fundamentally different from that due to Kelvin waves. While the velocity field induced by Kelvin waves is always zero in a direction perpendicular to the coast, or equator, it is in fact the displacement of particles perpendicular to the bottom contours that generates waves in a region with sloping bottom. We call these waves escarpment waves, and they arise as a consequence of the conservation of potential vorticity. Rigid lid The escarpment waves are essentially vorticity waves. The motion in these waves is a result of the conservation of potential vorticity. More precisely, the relative vorticity for a vertical fluid column changes periodically in time when the column is stretched or squeezed in a motion back and forth across the bottom contours. To study such waves in their purest form, we will assume that the surface elevation is zero at all times, i.e. we apply the rigid lid approximation. In this way the effect of gravity is eliminated from the problem. Let us assume that the bottom topography is as sketched in Fig. 1.13. Fig. 1.13. Bottom topography for escarpment waves. The continuity equation (1.1.20) can now be written as ( Hu ) x ( Hv ) y 0 . (1.8.1) 32 Accordingly, we can define a stream function satisfying Hu y , Hv x . (1.8.2) When linearizing, we obtain from the theorem of conservation of potential vorticity (1.1.25) that t f f u v 0 . H H x H y (1.8.3) We here assume that f is constant. Furthermore, we take that H H ( y ) . By inserting from (1.8.2), we can write (1.8.3) as 2 t H ( yt f x ) 0 , H (1.8.4) where H dH / dy . We assume a wave solution of the form F ( y ) exp( i(kx t )) . (1.8.5) By inserting into (1.8.4), this yields 2 kf H F k F 0 . 2 H H H (1.8.6) This equation has non-constant coefficients and is therefore problematic to solve for a general form of H(y). We shall not make any attempts to do so here. Instead, we derive solutions for two extreme types of bottom topography. One of these cases, where the bottom exhibits a weak exponential change in the y-direction, will be dealt with in section 10 in connection with topographic Rossby waves. The other extreme case, where the slope tends towards a step function, will be analysed here; see the sketch in Fig. 1.14. The escarpment waves relevant for this topography are often called double Kelvin waves. Fig. 1.14. The bottom configuration for double Kelvin waves. 33 For trapped waves, the solutions of (1.8.6) in areas (1) and (2) are, respectively F1 A1 exp( ky), F2 A2 exp( ky). (1.8.7) We note that these waves are trapped within a distance of one wavelength on each side of the step. At the step itself (y = 0), the volume flux in the y-direction must be continuous, i.e. v1H1 v2 H 2 , y 0. (1.8.8) This means that x (and thereby also ) must be continuous for y = 0, i.e. A1 = A2 = A in (1.8.7). Furthermore, the pressure in the fluid must be continuous for y = 0. The pressure is obtained from the linearized x-component of (1.1.1), i.e. px (ut fv) H ( Hu )t fHv . (1.8.9) Writing p P( y ) exp( i (kx t )) , and applying (1.8.2) and (1.8.5), we find that F f F P . H kH (1.8.10) By inserting from (1.8.7), with A1 = A2, into (1.8.10), continuity of the pressure at y = 0 yields the dispersion relation H H 1 f 2 . H H 1 2 (1.8.11) We note that we always have that f , and that the wave propagates with shallow water to the right in the northern hemisphere, i.e. > 0 when f > 0. These two properties are generally valid for escarpment waves, even though we have only shown it for double Kelvin waves with a rigid lid on top. In the case where the escarpment represents the transition between a continental shelf of finite width and the deep ocean, this type of waves are often called continental shelf waves. This kind of bottom topography is found outside the coast of Western Norway. Here, numerical results show the existence of continental shelf waves in the area close to the shelf-break, e.g. Martinsen, Gjevik and Røed (1979). The topographic trapping of long waves near the shelf-break and the currents associated with these waves, interact with the wind-generated surface waves, which tend to make the sea state here particularly rough. This is a well-known fact among fishermen and other sea travellers that frequent this region. 34 The effect of gravity In general, we must allow the sea surface to move vertically. Let us consider a wave solution of the form (u, v, ) exp( i (kx t )) . (1.8.12) For such waves, the linear versions of (1.1.18) and (1.1.19), with constant surface pressure, yield g k f y , f 2 ig kf y . v 2 f 2 u 2 (1.8.13) We write the surface elevation as G ( y ) exp( i (kx t )) . (1.8.14) Inserting into (1.1.20), we find 2 f 2 fk ( HG) k 2H H G 0 . g (1.8.15) For 2 f 2 , i.e. quasi-geostrophic motion, we revert to the gravity-modified escarpment wave. For f = 0 and H (tan ) y , this equation yields the so-called edge waves. These are long gravity waves trapped at the coast. The trapping in this case is not due to the earth’s rotation, but is caused by the fact that the local phase speed gH increases with increasing distance from the coast. If we represent the wave by a ray, which is directed along the local direction of energy propagation, i.e. perpendicular to the co-phase lines, the ray will always be gradually refracted towards the coast. At the coast, the wave is reflected, and the refraction process starts all over again. The total wave system thus consists of a superposition between an incident and a reflected wave in an area near the coast. The width of this area depends on the angle of incidence with the coast for the ray in question. Outside this region, the wave amplitude decreases exponentially. The edge waves can propagate in the positive as well as in the negative x-direction. The lowest possible wave frequency is given by 2 gk tan . (1.8.16) There are also a number of higher, discrete frequencies for this problem; see LeBlond and Mysak (1978), p. 221. If we take the earth’s rotation into account (f 0), the 35 frequencies for the edge waves in the positive and negative x-directions will be slightly different. 1.9 Planetary Rossby waves Like topographic waves, planetary Rossby waves are a result of the conservation of potential vorticity. However, in this case the relative vorticity of a vertical fluid column changes periodically in time when the planetary vorticity changes as the column moves back and forth in the north-south direction. We realize then, that this motion occurs on a beta-plane. The motion is essentially horizontal. As for escarpment waves, we can eliminate gravity completely from the problem by assuming that the surface is horizontal for all times, i.e. 0. The hydrostatic assumption then yields for the pressure p gz p( x, y, t ) . (1.9.1) The linearized equations thus become 1 vt fu p y , ( Hu ) x ( Hv ) y 0. ut fv 1 px , (1.9.2) First, we take that H is constant, and that f = f0+y. From the continuity equation, we can define a stream function such that u y , v x. (1.9.3) By eliminating the pressure from (1.9.2) we find 2H t x 0 , (1.9.4) where 2H 2 / x 2 2 / y 2 is the horizontal Laplacian operator. This equation also follows directly from the linearized potential vorticity conservation equation (1.8.3), when H is constant and fy = . We assume wave solutions of the form exp( i (kx ly t )) . (1.9.5) Inserting (1.9.5) into (1.9.4) yields for the frequency k k l2 2 . (1.9.6) 36 We shall discuss this dispersion relation in more detail, but first we repeat some elements of general wave kinematics. We introduce the wave number vector defined by k1i1 k 2 i2 k 3 i3 , and a radius vector r , where r r1i1 r2 i2 r3 i3 . (1.9.7) (1.9.8) A plane wave of the type (1.9.5) can now be written A exp( i ( r t )) A exp{i r t } . 2 Accordingly, the vectorial phase speed c can be defined by c 2 , 2 k12 k 22 k 32 . The components of the vectorial group velocity c g are given by cg(1) / k1 , cg( 2 ) / k2 , cg( 3) / k3 . (1.9.9) (1.9.10) (1.9.11) In vector notation this becomes c g , i1 i2 i3 . k1 k2 k3 (1.9.12) If the frequency only is a function of the magnitude of the wave number vector, i.e. ( ) , we refer to the system as isotropic. If we cannot write the dispersion relation in this way, the system is anisotropic. We now consider the surface in wave number space given by (k1 , k2 , k3 ) C , where C is a constant; see Fig. 1.15, where we display a two-dimensional example. Fig. 1.15. Constant- frequency surface in wave number space. 37 The gradient is always perpendicular to the constant frequency surface. From (1.9.12) we note that this means that the group velocity is always directed along the surface normal, as depicted in Fig. 1.15. Since the phase velocity is directed along the wave number vector, e.g. (1.9.10), we realize that if the phase speed and group velocity should become parallel, then the constant frequency surface must be a sphere in wave number space. Mathematically, this means that ( ) , i.e. we have an isotropic system. We now return to the discussion of the dispersion relation (1.9.6) for Rossby waves. Since (k , ) , the system is anisotropic. Hence, the group velocity and the vectorial phase speed are not parallel. Equations (1.9.6), (1.9.10) and (1.9.11) yield for the components of the phase speed and group velocity: c ( x) c ( y) k k2 2 4 , l kl 2 4 , (1.9.13) and (k 2 l 2 ) , k 4 2 kl 4 . l cg( x ) c ( y) g (1.9.14) We note that the x-component of the phase speed is always directed westwards (the xaxis is assumed to be parallel to the equator). However, the energy propagation can have an eastward or westward component, depending on the ratio l/k. For waves propagating along latitudinal circles (l = 0), the energy always propagates eastward. The magnitudes of the phase speed and the group velocity are easily obtained from (1.9.13) and (1.9.14): k c 2 , cg 2 . (1.9.15) We note that c | c g | . For a typical Rossby wave propagating along equator we take = 2/k = 1000 km. With = 2 10 11 m1s1, the phase speed and period become c 0.5 m s1 and T = |2/| 23 days, respectively. We realize that such Rossby waves are long-periodic phenomena. 38 Since is negative (we have chosen k > 0), we can define a positive quantity by: 1 . 2 (1.9.16) Equation (1.9.6) can then be written as k 2 l 2 2. (1.9.17) For a prescribed constant frequency , the wave number components k and l are obtained as points on a circle in the wave number plane. This circle has a radius and is centred at (, 0); see Fig. 1.16. The figure also depicts the phase speed and the group velocity obtained from (1.9.10) and (1.9.12). Fig. 1.16. Sketch of the relation between the phase speed and the group velocity for Rossby waves. We now consider the reflection of Rossby waves from straight coasts in the north-south direction. Here, we take that the east coast is situated at x = L, where the x-axis is parallel to the equator; see Fig. 1.17. Fig. 1.17. Reflection of Rossby waves from a straight coast. 39 As far as wave propagation is concerned, it is the group velocity that represents signal velocity. Accordingly, for a receiver, or a coast, to experience a physical impulse, it must have a component of the group velocity directed against it. Accordingly, the incoming wave has a group velocity with an eastward x-component. This wave is labelled with subscript 1, i.e. (1) A1 exp( i(k1x l1 y 1t )) . (1.9.18) The reflected wave, labelled with subscript 2, must have a westward-directed group velocity component. The reflected wave is written as ( 2) A2 exp( i(k2 x l2 y 2t )) . (1.9.19) At the east coast, the normal component of the velocity must vanish, i.e. y(1) y( 2) 0, x L , (1.9.20) or il1 A1 exp( ik1L) exp{i(l1 y 1t )} il2 A2 exp( ik 2 L) exp{i(l2 y 2t )} 0 . (1.9.21) If this is to be valid for all y and all t, we must have that l1 l2 , 1 2 , A1 exp( ik1L) A2 exp( ik 2 L). (1.9.22) Hence, the frequency and the wave number component in the y-direction are conserved during the reflection process. This means that the reflected wave number vector must describe the same circle as the incident one, with the same l; see Fig. 1.18. Fig. 1.18. Sketch of the relationship between wave characteristics of incident and reflected Rossby waves. 40 We notice that the group velocity satisfies the law of reflection in ray theory; that is the angle with respect to the normal of the reflecting plane is the same for the incident and for the reflected wave. We also note that an incident short wave is reflected from an east coast as a long wave, while reflection from a west coast (interchange subscripts 1 and 2 in Fig. 1.18), results in the transformation from a long wave to a short wave. For this reason the east coast is said to be a source of long waves, while the west coast is a source of short waves. Since short waves are subject to stronger dissipation than long waves (dissipation is proportional to the square of the velocity gradients), this may lead to an increased transition of mean momentum from waves to currents at the west coasts of the oceans. This has been used (Pedlosky, 1965) as basis for explaining why the ocean currents are more intense in such regions (western intensification). We shall return to that question in section 2. To see how the effect of a free surface modifies the Rossby wave that we have studied up to now, we consider the linearized version of the vorticity equation (1.1.22): t f (u x v y ) v 0 . (1.9.23) Eliminating the horizontal divergence u x v y in this equation by using the linearized form of (1.1.20), we find (v x u y ) t f t v 0 . H (1.9.24) We now simplify the problem by assuming that the time scale for the Rossby wave is so large (several days) that the pressure gradient force and the Coriolis force always have time to adjust to a state of geostrophic balance. This means that the velocity field in (1.9.24) can be approximated by v g x / f , u g y / f , (1.9.25) where f is a mean (constant) value of f. This relation is often called the quasigeostrophic approximation and requires that 2 f 2 . By replacing f with f in (1.9.24) and inserting from (1.9.25), we obtain 2Ht x a 2t 0 , where a ( gH )1 / 2 / f . By assuming solutions of the form (1.9.26) 41 exp( i(kx ly t )) , (1.9.27) the dispersion relation is obtained from (1.9.26): k k l 2 a 2 2 . (1.9.28) If the wavelength is much smaller than the Rossby radius (k 2 , l 2 a 2 ) , the effect of the surface elevation can be neglected, and we are back to our rigid lid case (1.9.6). However, if the wavelength is large (k 2 , l 2 a 2 ) , we find from (1.9.10) that c( x) c( y) k 2a 2 k2 l2 kla 2 k l 2 2 k2 ( a ) 2 , 4 (1.9.29) kl ( a ) 2 . 4 (1.9.30) In this case, the group velocity components become cg( x ) a 2 , c g( y ) 2 kl 4 ( a ) 4 . (1.9.31) (1.9.32) Since a now is a small parameter, we realize that the energy mainly propagates westward in this case. If the wave number vector is parallel to the equator (l = 0) we have non-dispersive waves, i.e. c ( x ) cg( x ) a 2 , and c ( y ) cg( y ) 0 . 1.10 Topographic Rossby waves Let us assume that somewhere the relative vorticity is zero. From the theorem of conservation of potential vorticity (1.1.25) with 0, we find that a displacement northwards, where f is increasing, generates negative (anti-cyclonic) relative vorticity. However, we realize that the same effect can be achieved by a northward displacement if f is constant and the depth H decreases northward. This gives rise to the so-called topographic Rossby waves. Of course, the existence of such waves does not require that the bottom does slope in one particular direction. Topographic Rossby waves are only a special case of escarpment waves when the bottom has a very weak exponential slope everywhere in the ocean. By letting H H 0 exp( y) , equation (1.8.6) reduces to fk F F k 2 F 0 . By assuming (1.10.1) 42 F A exp( iy ) , (1.10.2) insertion into (1.10.1) yields the complex dispersion relation i k 2 2 fk 0. (1.10.3) In general we may allow for a very weak change of wave amplitude in the direction normal to the coast, i.e. we take in (1.10.2) to be complex: l i . (1.10.4) By insertion into (1.10.3), the imaginary part leads to = /2 (when l 0). From the real part of (1.10.3) we then obtain fk . k l2 2 / 4 2 (1.10.5) We note that the along-shore phase speed c ( x ) / k is positive. This means that the wave propagation in the along-shore direction is such that we have shallow water to the right (in the northern hemisphere). For a bottom that slopes gently compared to the wavelength (k >> ), we see from the (1.10.5) that these waves are similar to short planetary waves propagating in an ocean of constant depth. The expressions for the frequency are identical, e.g. (1.9.6), if f . (1.10.6) This similarity is often used in laboratory experiments in order to simulate planetary effects. When l = 0, the equations are satisfied for = 0, i.e. constant amplitude waves. Such waves propagate along the bottom contours with shallow water to the right, and mimic planetary Rossby waves along latitudinal circles in an ocean of constant depth. 1.11 Barotropic instability Rossby waves in the ocean can be generated by atmospheric forcing, as will be shown in section 2. They can also develop due to energy transfer from unstable ocean currents. Assume that we have a stationary mean flow along a latitudinal circle (1.11.1) v U ( y )i , and that this initial state is in geostrophic balance, i.e. Py f U . We disturb (perturbate) the initial state such that the velocities and the pressure can be written 43 u U u~( x, y, t ), v v~( x, y, t ), pP ~ p ( x, y, t ). (1.11.2) Here we have assumed that the motion is two-dimensional, i.e. independent of z. If the perturbations (denoted by ~) are sufficiently small, the equations governing the motion can be linearized. With w = 0, i.e. constant depth and horizontal surface in the model depicted in Fig. 1.1, equations (1.1.1) and (1.1.3) reduce to ~ py , 1 u~t U u~x v~U f v~ ~ px , 1 v~t U v~x f u~ u~x v~y 0. (1.11.3) By introducing the stream function for the perturbation, defined by u~ , v~ , and eliminating the pressure from the equation above, we find y x 2H t U ( y ) 2H x ( U ) x 0 . (1.11.4) For U 0, (1.11.4) reduces to the equation for Rossby waves (1.9.4). We assume that the solutions can be written as ( y ) exp( ik ( x ct )) . (1.11.5) Furthermore, the ocean is restricted by two straight coasts, defining a channel in the xdirection; see Fig. 1.19. Fig. 1.19. Horizontal shear flow in an ocean with parallel boundaries. 44 At y = 0 and y = D the normal velocity must be zero, i.e. x = 0. Since this is valid for all x along the boundaries, we must accordingly have that = 0 at y = 0, D. Insertion from (1.11.5) into (1.11.4) yields U k 2 0 , U c (1.11.6) with boundary conditions 0, y 0, D . (1.11.7) If 0, equation (1.11.6) is reduced to the well-known Rayleigh equation for twodimensional perturbations of shear flow in a homogeneous, inviscid fluid. We assume that the wave number k in (1.11.5) is real, while the phase speed c and the amplitude function are generally complex, i.e. c cr ici , r i i . (1.11.8) From (1.11.5) we see that if ci > 0, will grow exponentially in time. The initial state is then said to be unstable, and we have an energy transfer from the mean flow to the disturbance, which in this case is a Rossby wave. Since equation (1.11.6) for the perturbation amplitude has non-constant coefficients and a singularity for U = c (the term containing the highest derivative vanishes at locations in the fluid where U = c), it is in general quite complicated to solve. We will not attempt to do so here. Instead of a detailed investigation of the stability conditions, we will be content with the derivation of a necessary condition for the occurrence of instability. First, we make the denominator in (1.11.6) real, i.e. U U cr ici k 2 0 . 2 2 U cr ci (1.11.9) Now we multiply this equation with the complex conjugate amplitude function r ii , and note that r2 i2 0 . By integrating the resulting 2 equation from y = 0 to y = D, and applying the boundary conditions 0 at y = 0, D, we find that 2 2 D 2 k 2 2 U U cr dy ic U dy 0 . i 2 0 2 U cr 2 ci2 0 U cr ci D (1.11.10) For this equation to be satisfied, both real and imaginary parts must be zero. It suffices here to consider only the imaginary part, i.e. 45 U 2 dy 0 . 2 2 U c c r i 0 D ci (1.11.11) If ci > 0, i.e. the basic flow is unstable, the integral in (1.11.11) must be zero. However, the integrand is positive, possibly except for the factor U in the numerator. Hence, for the integral to become zero, U must change sign in the fluid, which means that there must be at least one place where U . This expresses a necessary (but not sufficient) condition for instability of the basic flow. An equivalent statement is that the absolute vorticity of the basic flow, abs f U , must have an extreme somewhere in the ocean (where d abs / dy 0 ). This kind of instability is related to the energy transfer from the kinetic energy of the mean flow (via the velocity shear) to the perturbations, and is called barotropic instability. Perhaps even more common in the ocean is baroclinic instability, where energy is transferred from the potential energy of the basic state to the perturbations via the density stratification. We will return to discuss baroclinic instability in section 3. Finally, we mention that on an f-plane ( = 0), the necessary condition for instability is that the basic flow profile must have a point of inflexion ( U 0 ) somewhere in the fluid. Also, the numerical value of the vorticity U must have a maximum where U 0 (the Rayleigh-Fjørtoft criterion). The last condition can be found by applying that the real part of (1.11.10) is zero, together with (1.11.11) and ci 0. 1.12 Barotropic flow over an ocean ridge We consider an ocean current that crosses a sub-sea ridge. For simplicity, we assume that the ridge is infinitely long in the y-direction, and that the depth far away from the ridge is constant (= H0); see Fig. 1.20, 46 Fig. 1.20. Model sketch. The surface far upstream is horizontal and the constant upstream flow U0 is geostrophic, i.e. fU 0 1 Ps . y (1.12.1) On an f-plane, which we assume here, this means that the surface pressure Ps of the basic flow must vary linearly with y. We study the effect of the ridge by applying the potential vorticity theorem (1.1.25). In a steady state we have v Q 0 . (1.12.2) A possible solution of this equation is Q const. , in the (x, y)-plane. We assume that the effect of the ridge is not felt far upstream, i.e. Q f / H 0 for all values of y when x . Since Q is a conserved quantity, we then must have Q f f H H0 (1.12.3) We introduce the shape of the ridge q(x), i.e. H H 0 q ( x) . (1.12.4) The typical width of the ridge is 2L; see Fig. 1.20. Inserting (1.12.4) into (1.12.3), and assuming that / y 0 , we obtain vx f ( q) . H0 (1.12.5) 47 In the literature one often finds discussions of this problem where the surface is modeled as a rigid lid, i.e. 0 . In this case we notice from (1.12.5) that anticyclonic vorticity is generated over the entire ridge. By integration we obtain x v f q ( )d . H 0 L (1.12.6) For x L , we find that v = 0. For x L , we obtain v v ( ) fA , H0 (1.12.7) where A is the cross-section of the ridge, given by A L L q( )d q( )d . (1.12.8) The corresponding streamlines are sketched in Fig. 1.21. Fig. 1.21. Deflection of flow over a ridge when the surface is approximated by a rigid lid. The deflection angle of the streamlines on the lee side of the ridge is v ( ) fA arctan . U0 U0H0 arctan (1.12.9) The solution (1.12.6) may not be realistic, since it (through geostrophic balance in the x-direction) implies that p / x must be constant far downstream. This means that the perturbation pressure becomes infinitely large when x . However, the solution obtained here may be of interest close to the ridge. In most geophysical applications the surface can move freely in the vertical direction. This means that a fluid column can be stretched, and not just be equal to, or 48 shorter than the upstream value, as in our previous example. From the linearized versions of (1.1.18)-(1.1.19), we obtain for the surface elevation that is induced by the ridge: x U 02 f v xx v . gf g (1.12.10) By insertion into (1.12.5), we find (1 F02 )v xx 1 f v qx . 2 a H0 (1.12.11) Here F0 U 0 /( gH 0 )1 / 2 is the Froude number, and a ( gH 0 )1 / 2 / f is the Rossby radius. In the ocean F0 << 1, while we typically have that a 103 km. It is then reasonable to assume that L << a. Hence, on the left-hand side of (1.12.11), the first term dominates. Accordingly, a particular solution, when F0 << 1, can be written v( p) f qdx C1 x C 2 . H0 (1.12.12) The solution of the homogeneous equation is: v ( h ) C3 exp( x / a) C4 exp( x / a ) . (1.12.13) The coefficients C1, C2, C3, and C4 are determined from the boundary conditions. We assume that the ridge is symmetric about x = 0, and we require that the velocity and the pressure are continuous at x = 0. Furthermore, we assume that v 0, x , and that the perturbation pressure becomes finite when x . The last condition is equivalent to require vanishing v at infinity (in the case of no waves behind the ridge). The solution becomes: x fA f exp( x / a) q( )d , x 0, H 0 2H 0 v x fA {1 exp( x / a)} f q( )d , x 0. 2H H 0 0 0 (1.12.14) Here A is the cross-sectional area defined by (1.12.8). This solution yields v (0) 0 . For x < L, we find that v fAexp( x / a ) /( 2H 0 ) 0 , while for x > L we have that v fAexp( x / a ) /( 2 H 0 ) 0 . For a symmetrical ridge, v becomes anti-symmetric about the centerline of the ridge. The streamlines in this case are sketched in Fig. 1.22. 49 Fig. 1.22. Deflection of flow over a symmetric ridge when the surface is free to move. In this case we only have a local disturbance of the current in the area close to the ridge. This is also obvious from the conservation of potential vorticity (1.12.3): f f . H0 q H0 (1.12.15) We have stretching of a fluid column, and an associated generation of cyclonic vorticity just before, and just after the ridge. Over the higher parts of the ridge the column becomes compressed, and here anti-cyclonic vorticity is generated; see the schematic sketch in Fig. 1.23. Fig. 1.23. Conservation of potential vorticity for a fluid column when the upstream fluid depth H0 is independent of the y-coordinate. In the present case we will always have an elevation of the surface over the ridge, as sketched in Fig. 1.23. By inserting from (1.12.5) into (1.12.14), we find that ( 0) A 0. 2a (1.12.16) 50 The elevation computed here is caused by the effect of the earth’s rotation. In the general case where the Froude number is not small, we find from (1.12.5) and (1.12.10) that xx F02 1 1 q 2 q. 2 2 2 xx a (1 F0 ) 1 F0 a (1 F02 ) (1.12.17) Solutions of this equation may yield surface elevations or surface depressions above the ridge, depending of the form of q, and the magnitudes of a and F0. We will not discuss the general solution here. However, for a non-rotating fluid (f = 0, i.e. a), we find from (1.12.17) that F02 q( x ) . 1 F02 (1.12.18) This is a well-known result in fluid mechanics. It states that for sub-critical currents (F0 < 1), we have (0) 0 above a ridge. For super-critical currents (F0 > 1), we find (0) 0 . Examples of both these cases are found for flow over bottom rocks in shallow rivers. Stationary Rossby waves behind ridges For large-scale motion over wide ridges we must take into account the fact that the Coriolis parameter varies with the latitude, i.e. f f 0 y . Then stationary, or standing, planetary Rossby waves can be generated by the presence of the ridge. In general, the condition for the existence of a standing wave is that it must have a phase speed that is equal in magnitude, and oppositely directed to the basic current. The direction of the energy flow (basic current plus group velocity), then decides on which side of the ridge we find the standing wave, because non-damped wave motion cannot exist without continuous supply of energy to the wave region. In our case, the phase of the Rossby wave moves westward, while the energy moves eastward. This apply to planetary Rossby waves where the wavelength and the Rossby radius a satisfy the relation 2 a ; see the dispersion relation (1.9.28). We then realize immediately that standing Rossby waves may occur on the lee-side of the ridge (here: for x > 0), if the current is directed from the west towards the east. Furthermore, it is obvious that we cannot have standing Rossby waves at all when the current is directed towards the west. In the atmosphere, standing, mountain-generated Rossby waves in 51 the westerlies may be responsible for observed distributions of pressure ridges and pressure troughs; see Charney and Eliassen (1949). Alternatively, on an f-plane, we may find standing topographic Rossby waves on the lee-side of the ridge in our example (Fig. 1.20) if the ocean bottom slopes upwards in the y-direction (remember: decreasing H gives the same effect as increasing f in the potential vorticity, e.g. sec. 1.10). If H H 0 exp( y) , where < 0, the frequency for short Rossby waves is f / k 0 , e.g. (1.10.5). As mentioned before, a standing wave requires a phase speed that is equal, and oppositely directed to the basic flow, i.e. k U0 . (1.12.19) Accordingly, the wavelength of standing topographic Rossby waves becomes U 2 0 f 1/ 2 . (1.12.20) This scale is relevant for laboratory simulations of Rossby waves in rotating tank experiments. When ocean currents are flowing over isolated sub-sea mountains or banks, anticyclonic vorticity may be generated over the top of the mount. If the vorticity is sufficiently strong, as it can be for high mounts, a permanent vortex, or a Taylor column (Taylor, 1922) may be formed. Measurements from the Halten Bank west of Norway (Eide, 1979), suggest the presence of such vortexes, or eddies. This means that the water mass above the bank may have a rather long residential time. But bank regions are important for the early growth of fish larvae. Therefore, in the case of accidental oil spills, or other kind of pollution, the long residence time of the water mass means that the fish larvae will stay in contact with the polluted water for quite some time. 2. WIND-DRIVEN CURRENTS AND OCEAN CIRCULATION 2.1 Equations for the mean motion Mass and momentum The velocity introduced in Section 1 was related to the displacement of individual 52 fluid particles. However, many oceanic flow phenomena are turbulent, which means that it is difficult to keep track of individual particles. But turbulent flows are not without structure. Even if individual particles move in a chaotic way, they may in an average sense define an orderly flow. In the ideal case the mean flow has a much larger characteristic time scale than the rapid fluctuations of individual fluid particles. Then we can define mean velocities by a time averaging process: 1 v T t T / 2 v dt . (2.1.1) t T / 2 Here T represents a period that is large compared to the characteristic scale of the turbulence, but small compared to the typical time scale of the mean motion we intend to study. The same type of averaging must be done for the pressure and the density. Accordingly we write v v v , , p p p. (2.1.2) where v , , p represent the fluctuating (turbulent) part of the motion. The continuity equation (1.1.2) expresses the conservation of mass in a fluid, and is a fundamental law in non-relativistic mechanics. Obviously, molecular processes like heat and salt diffusion may change the density of a fluid particle. This effect is accounted for by the application of the equation of state for the fluid. The change of density has a very important dynamical implication in that it changes the buoyancy of a fluid particle, which in turn may lead to thermohaline circulations in the system. Equation (1.1.2) is derived under the assumption that only advective fluxes (transport of mass by the velocity field) can change the mass within a fixed, small geometrical volume. This assumption is essential for the definition of the velocity field in the fluid. It is a fact that for most low-frequency phenomena in the ocean, the individual change of density for a fluid particle is very small, and can be neglected in the continuity equation. Hence, we take that v 0, (2.1.3) i.e. the velocity field in the ocean is approximately non-divergent. By averaging this equation and applying (2.1.2), we find that 53 v 0, v 0. (2.1.4) By separating the variables as in (2.1.2), and applying (2.1.4) we find from averaging the viscous Navier-Stokes equation in a reference system rotating with constant angular velocity f/2 about the vertical axis, that 1 vt v v fk v p gk v v v . (2.1.5) We have neglected compared to in the pressure term, since typically in the ocean / ~ 103 . Furthermore, is the kinematic molecular viscosity coefficient ( 102 cm2s1). The term v is usually much smaller than the turbulent momentum transport per unit mass, v v , and can therefore be neglected. The term v v is usually referred to as the turbulent Reynolds stress tensor. We first consider the case of isotropic turbulence, i.e. turbulence that has no preferred directions, and introduce a constant kinematic eddy viscosity A0. Analogous to the stress tensor for a Newtonian fluid, we assume for the turbulent Reynolds stress tensor per unit density: v v j 2 viv j A0 i ij K , i, j 1,2,3 . x j xi 3 (2.1.6) Here 1, i j , 0, i j , ij and K 1 3 vivi . 2 i 1 The quantity K is the kinematic energy per unit mass of the turbulent fluctuations. The sum of the diagonal, turbulent Reynolds stress components (i = j) must always equal 2K. We note from the right-hand side of (2.1.6) that this is true for our choice of parameterisation, since v 0 . The term 2 K / 3 formally acts as an “isotropic” pressure. This term can be combined with the mean pressure. In the momentum equation we then write 1 2 1 2 1 p K ( p K ) p* , 3 3 (2.1.7) neglecting the variation of in connection with K. If we assume that A0 is much larger than the molecular viscosity coefficient, insertion from (2.1.6) and (2.1.7) into (1.2.5) yields 54 1 vt v v fk v p* A0 2v gk . (2.1.8) Here 2 2 / x 2 2 / y 2 2 / z 2 is the three-dimensional Laplacian operator. For large-scale ocean circulation it is usual to assume that the turbulence is dependent on direction, and we define AH and A as turbulent eddy viscosity coefficients in the horizontal and the vertical directions, respectively. The surfaces of constant density are mainly horizontal in the ocean, and the stratification is stable. Hence the turbulence activity in the vertical direction will be smaller than in the horizontal direction. Therefore one usually assumes that AH >> A. For such motion wz is often negligible, i.e. u x v y 0 . Furthermore, we typically have u2 v 2 w2 . Accordingly, we can assume for the normal stresses: uu 2 AH u x K , vv 2 AH v y K , ww 0, (2.1.9) while the tangential stresses become: uv vu AH (vx u y ), uw wu Au z AH wx , vw wv Avz AH wy . (2.1.10) Furthermore, we assume that AH is constant, while we permit that A A(z ) . The horizontal component of (2.1.5) can then be written 1 qt v q fk q H p* AH 2H q ( Aqz ) z , (2.1.11) where q ui vj , 2H 2 / x 2 2 / y 2 , and p* p K . For large-scale ocean circulation we can assume that the vertical pressure distribution is approximately hydrostatic. This means that we can replace the zcomponent in (2.1.5) by p g dz PS ( x, y, t ) , (2.1.12) z where PS is the atmospheric pressure at the sea surface. Heat and salt The internal energy per unit mass of sea water, e, can be written approximately as 55 e cv T , (2.1.13) where cv is the specific heat at constant volume, and T the absolute temperature. By considering a geometrically fixed volume with surface , we find for the change of internal energy with time within that (2.1.14) e ( e v ) F T , t where the vectorial area is normal to the surface and points outwards from the considered fluid volume. The two terms on the right-hand side represent the advective and diffusive surface heat fluxes, respectively, into the volume. We assume that the diffusive energy flux, FT , is given by Fourier's law: FT kT T . (2.1.15) Here kT is the coefficient of heat conduction. Applying Gauss’ theorem, we obtain from (2.1.14) t ( c T ) ( c T v ) (k T ) 0 . v v T (2.1.16) This equation is valid for an arbitrary volume in the fluid. Accordingly, the integrand must be zero everywhere. Furthermore, by taking that cv and kT are constants, which are reasonable assumptions, and using (2.1.3), we find from (2.1.16): T v T T 2 T . t (2.1.17) Here T kT /( cv ) is called the thermal diffusivity. Equation (2.1.17) is often referred to as the heat equation. Equivalently, the change of salt with time within a geometrically fixed volume is given by S ( S v ) F S , t (2.1.18) where the terms on the right-hand side represent the advective and diffusitive surface salt fluxes, respectively, into the fluid volume. We assume that the diffusitive salt flux, FS , is given by Fick’s law (2.1.19) FS k S S , where kS is the diffusion coefficient for salt in water. As for heat, we now obtain for salt from (2.1.18) 56 t ( S ) ( Sv ) (k S ) 0 . (2.1.20) S If we assume that and kS are approximately constants in this equation, and apply (2.1.3), we find that S v S S 2 S . t (2.1.21) Here S k S / is called the salt diffusivity. Equation (2.1.21) is often referred to as the salt equation. The diffusion coefficients T and S are determined by the molecular structure of the fluid. They vary in fact with the temperature and the salinity, but this variation is small, and we will regard them as constants. Typical values for seawater are T 103cm2s1 and S 105cm2s1. We define T T T and S S S . Furthermore, we introduce turbulent eddy diffusivities K(T) and K(S) for temperature and salt. The eddy diffusivities are generally much larger than the molecular diffusivities in (2.1.17) and (2.1.21). Analogous to the molecular fluxes from Fourier’s and Fick’s laws, we now assume for the turbulent transports uT K H(T )Tx , vT K H(T )Ty , wT K (T )Tz , uS K H( S ) S x , vS K H( S ) S y , wS K ( S ) S z . (2.1.22) As before, subscript H denotes horizontal direction. We assume that the horizontal eddy diffusivities are constant. Hence the heat and salt equations become, when we neglect the effect of molecular diffusion: Tt v T K H(T ) 2H T ( K (T )Tz ) z , (2.1.23) S t v S K H( S ) 2H S ( K ( S ) S z ) z . (2.1.24) In fully developed turbulent motion the exchange of heat and the exchange of salt must be approximately similar, i.e. K H(T ) K H( S ) , K (T ) K ( S ) . (2.1.25) However, the mechanisms for momentum exchange in turbulent flows may be different from those related to heat/salt, so that we generally have AH KH and A K. In the upper mixed layer in oceans and fjords the motion sometimes seems to be quasi-turbulent. In such cases we find that K(T) > K(S) (Aas, private communication). 57 To get a closed system, the equation of state must be specified. The general form is F (T , S , p) . (2.1.26) For seawater, F is a very complicated function. Usually we can replace the variables in F with its mean values, i.e. F T , S , p . (2.1.27) We have now seven equations, (2.1.4), (2.1.11), (2.1.12), (2.1.23), (2.1.24), (2.1.27) for the seven unknowns, u , v , w , p, , T , S , which vary in space and time. When we in the following use some of these equations to describe the mean motion, the mean sign will be left out for simplicity. We also replace p* by p , i.e. we neglect the contribution from the turbulent Reynolds stresses to the mean pressure. Along the sea surface we have wind stress components S( x ) , S( y ) and a variable air pressure PS ( x, y, t ) that act on the ocean. The wind stress components are related to the wind speed (U10, V10) at 10 meters through the semi-empirical relations S( x ) a cD (U102 V102 )1 / 2U10 , S( y ) a cD (U102 V102 )1 / 2V10. (2.1.28) where a and cD are the density of air (1.2103g cm3) and the drag coefficient, respectively. Attempts have been made to assess the value of cD from measurements, and the results vary considerably. Values between 1103 and 3103 are reported in the literature, where higher values correspond to increasing wind speeds. 2.2 The Ekman elementary current system We consider stationary motion in a quasi-homogeneous ocean. By quasihomogeneous we mean that the generally varying density in the ocean can be approximated by a constant value in these studies. The ocean extends to infinity in the horizontal directions, and has constant depth H. We assume that the ratio between the convective acceleration and the Coriolis force, defining the Rossby number, R, is small, i.e. R v v H / fk v H 1 . (2.2.1) This allows us to disregard the acceleration term in the momentum equation. We also assume that horizontal turbulent diffusion of momentum can be neglected. Equation (2.1.11) for the mean variables (skipping the over-bar) then reduces to 58 px ( Auz ) z , 1 fu p y ( Avz ) z . fv 1 (2.2.2) By assuming constant pressure gradients and constant A in the entire fluid, (2.2.2) has an exact solution (e.g. Krauss, 1973, p. 248). However, friction is only important in relatively thin layers close to the surface and the bottom. These are the so-called Ekman layers, with thickness S and B, respectively. In the following we let the indices S and B signify the upper and lower Ekman layer, respectively. In these layers we assume that the eddy viscosities AS and AB are constant, but differing in numerical value. Furthermore, we assume that H >> S, B, implying that we have a large interior where the effect of friction can be completely neglected. Here we have geostrophic balance. Assuming that the horizontal pressure gradient is known, i.e. H p g H H PS from (2.1.12), the geostrophic velocity v G is given by px , 1 fuG p y . fvG 1 (2.2.3) We assume here that f is constant. Then we note from (2.2.3) that the geostrophic flow is non-divergent: u xG v Gy 0 . (2.2.4) The total steady flow is subdivided into an Ekman part and a geostrophic part: (2.2.5) v v E ( x, y, z ) v G ( x, y) . At the surface the tangential stress in the fluid must equal the wind stress given by (2.1.28). This leads to u S( x ) , z v AS S( y ) . z AS z 0. (2.2.6) In general, we will allow for a slipping motion at the bottom. For simplicity we assume that the frictional stress at the bottom is proportional to the bottom velocity (Rayleigh friction). Hence 59 u B r uB , z z H , (2.2.7) vB ( y) B AB r vB , z where r is a friction coefficient and v B v BE v BG . Formally, (2.2.7) yields the extreme conditions “no-slip”, or vB 0 in the limit r , and “free-slip”, or vB / z 0 when r = 0. B( x ) AB We start out by studying the Ekman layer at the bottom, and we define a boundary-layer coordinate ~z such that ~ z ( H z) / B , (2.2.8) where B (2 AB / f )1 / 2 is the thickness of the Ekman layer at the bottom. By introducing the complex velocity W uBE iv BE , the equation for the Ekman current in the bottom layer becomes W~~ 2 2 iW 0 . zz (2.2.9) The boundary conditions at bottom can be written AB (uBE u G ) E G u u , B ~ z r B E G AB (vB v ) E G vB v . ~ z r B ~ z 0, (2.2.10) Furthermore, above the bottom layer, we must have that v B v G . Accordingly, in our new coordinate: v BE 0, ~ z . (2.2.11) A solution of (2.2.9) that satisfies (2.2.11), can be written W C exp{ (1 i )~ z} , (2.2.12) where C = C r+ iCi is complex. By insertion into (2.2.10), we find 1 ( K 1)u G KvG 2K 2K 1 1 Ci Ku G ( K 1)v G , 2 2K 2K 1 Cr 2 , (2.2.13) where K ( AB f / 2)1 / 2 / r . Now, the total velocity in the bottom Ekman layer can be written uB u G Cr cos ~ z Ci sin ~ z exp( ~ z ), G ~ ~ ~ vB v Ci cos z Cr sin z exp( z ). (2.2.14) 60 For simplicity, we take that u G 0 , and let r (or K0), i.e. we apply a no-slip condition at the bottom. We then find u B v G sin ~ z exp( ~ z ), vB v G (1 cos ~ z exp( ~ z )). (2.2.15) From this result we see that the flow in the bottom Ekman layer (the Ekman spiral) is veering to left of the direction of the geostrophic current as we move towards the bottom. For the other extreme case at the bottom, when K (free slip), we find that Cr = Ci = 0, and accordingly v B v G . We realize that we need a frictional force at the bottom to obtain an Ekman spiral. By choosing AB = 10 cm2s1 and AB = 100 cm2s1 as examples, we find that B becomes 14 m and 44 m, respectively. With r = 0.24 cm s1, which often is used for shallow oceans, we find that K = 0.1 and 0.3. It could be realistic to expect that K 0.3 in the ocean. Taylor (1916) solved the analogous problem for the atmosphere. He assumed that the frictional stress at the ground was proportional to the square of the wind speed, instead of using the simplified model with linear friction, as assumed here. In the upper Ekman layer we define a boundary-layer coordinate z z / S , (2.2.16) where S (2 AS / f )1 / 2 . By introducing the complex velocity W uSE iv SE for the Ekman flow, we now find that Wz z 2 2iW 0 . (2.2.17) The boundary conditions (2.2.6) can now be written uSE T ( x) , z vSE T ( y) , z z 0, (2.2.18) where T ( x) 2 AS f 1/ 2 2 S( x ) , T ( y ) AS f 1/ 2 S( y ) . (2.2.19) We have assumed that H >> S. Accordingly, for the Ekman current u SE , v SE 0, z . (2.2.20) The equations (2.2.17)-(2.2.20) define the traditional Ekman spiral for a deep ocean. The solution is 61 1 (T ( x ) T ( y ) ) cos z (T ( x ) T ( y ) ) sin z exp( z), 2 1 E vS (T ( x ) T ( y ) ) cos z (T ( x ) T ( y ) ) sin z exp( z). 2 uSE (2.2.21) This solution describes the oceanic surface Ekman spiral veering to the right (in the northern hemisphere) compared to the wind direction. The surface current is deflected 45 to the right of the wind. The total current in the surface layer becomes uS uSE u G , vS vSE vG . (2.2.22) The oceanic flow described in this section is driven by wind stresses and pressure gradients, and is referred to as Ekman’s elementary current system. The pressure gradients induce a geostrophic flow, which is felt from the top to the bottom (independent of z). Close to the ocean bottom this flow is modified by friction in the Ekman layer. As result, the flow here is veering to the left of the geostrophic flow direction. The wind stress is only felt in the surface Ekman layer, inducing a current that is veering to the right of the wind direction. 2.3 The Ekman transport The volume transport in the lower Ekman layer is defined by ~ u dz u d z , B B B H 0 H B 1 VB vB dz B vB d~ z, H 0 UB H B 1 (2.3.1) where ~z is defined by (2.2.8). As before, we take u G 0 . By inserting from (2.2.14) into (2.3.1), we find 2K 1 BvG , 2 2 (2 K 2 K 1) 1 G VB 1 Bv , 2 2 ( 2 K 2 K 1 ) UB (2.3.2) where we have utilized that exp( ) 1 . The volume transport is deflected to the left of the geostrophic flow in the interior. We define a deflection angle B as B arctan UB . VB (2.3.3) For K = 0 (no-slip), B = 11.3, while for K = 0.3 (corresponding to AB = 100 cm2s1), 62 we find B = 9.4. Therefore, it is reasonable to assume that the volume transport in the bottom Ekman layer is deflected about 10 to the left of the geostrophic flow. We note that this result is derived by using a constant AB in the Ekman layer. In the upper layer, the Ekman transport is independent of the variation of the vertical eddy viscosity. Applying constant pressure in (2.2.2) and integrating over the boundary layer, while using the boundary conditions (2.2.6), we find that dz S( y ) /( f ), S 0 E E ( x) VS vS dz S /( f ). S 0 U E S u E S (2.3.4) We notice the well-known result that the Ekman transport in the surface layer is deflected 90 to the right of the wind direction (in the northern hemisphere). The result (2.3.4) is based on the fact that the viscous stresses in the fluid are approximately zero for z S, which can be seen to be true from (2.2.21). However, we must remember that (2.2.21) was derived by assuming a constant eddy viscosity in the upper layer, while this is not done for the Ekman transport. When the eddy viscosity varies in the upper layer, a precise definition of S in (2.3.4) is not entirely obvious. However, we may take that S (2 AS / f )1 / 2 , where AS is some kind of average (bulk) value for the upper layer. 2.4 Divergent Ekman transport and forced vertical motion (Ekman suction) We assume that the geostrophic flow and the wind stress vary slowly in space and time. By integrating the continuity equation (2.1.4) for the mean flow vertically across the bottom Ekman layer, we find wB U B VB , x y (2.4.1) where wB is the vertical velocity at z H B . Furthermore, we have utilized that w(H) = 0 (horizontal bottom). By inserting (2.2.14) into (2.3.1), we can calculate UB and VB. To simplify, we assume that K = 0. Applying (2.2.4), we then finally obtain from (2.4.1) wB B k ( v G ) B G , 2 2 (2.4.2) where G is the geostrophic vorticity, and we have neglected exp() in comparison 63 with 1. A similar integration of the continuity equation in the upper Ekman layer yields wS U S VS 1 k ( S ) , x y f (2.4.3) where the final result follows from (2.3.4). Here wS is the vertical velocity at z B . Furthermore, we have neglected any vertical motion of the sea surface, i.e. we have taken w(0) = 0. The situation is sketched in Fig. 2.1. Fig. 2.1. Schematic model of Ekman suction. As an example, we let a wind field with maximum wind speed of 40 knots ( 20 m s1) vary over a typical distance L = 100 km. From (2.1.28) we then estimate that S /(L) ~ 106cm s2. With f = 104s1, equation (2.4.3) yields wS ~ 0.01cm s1. This, in turn, means that this wind field must be “turned on” for 11½ days to transport a particle 100 m vertically. In comparison, we can estimate the turbulent vertical diffusion velocity wturb. For a typical vertical length scale H and a time scale T, the diffusive terms balance; see for example (2.1.11), when H/T ~ A/H. This defines a turbulent diffusion velocity scale, i.e. wturb ~ A/H. With A = 100 cm2s1 and H = 100 m, we find that wturb ~ 102 cm s1, which is the same order of magnitude as for the forced vertical suction velocity in this example. Let us now assume that the wind stress is zero. Then wS = 0 from (2.4.3). In the interior of the fluid, where we can disregard the effect of friction, the linearized vorticity equation from (1.1.18)-(1.1.19) for non-divergent flow on an f-plane reduces to t f wz . Vertical integration in the interior yields (2.4.4) 64 S dz f w t B , (2.4.5) H B since we have assumed that wS = 0. Because of the vertical transport of fluid particles in the interior, the geostrophic vorticity in this region changes slowly in time, i.e. G F (t ) ( x, y) . (2.4.6) We assume that t tG Ft in equation (2.4.5). By utilizing that H >> S, B, we approximately have from (2.4.5): H tG f wB f B G , 2 (2.4.7) where the last result follows from (2.4.2). Inserting from (2.4.6): Ft f B F. 2H (2.4.8) This equation has a simple exponential solution. The slowly-changing geostrophic vorticity can then be written G ( x, y ) exp( f B t) . 2H (2.4.9) We notice that the vorticity decreases in time. This phenomenon is referred to as spindown, and is due to the vertical transport of fluid particles caused by divergence in the bottom Ekman layer. In this way the friction, which only is important in a relatively thin region close to the bottom, may effectively influence fluid motion in the interior. If we take AB = 10 cm2s1 and H = 100 m, we find a spin-down time T = 2H/(fB) 6 days from (2.4.9) With A = 10 cm2s1 in the entire fluid, it will take about T ~ H2/A ~ 100 days before the friction in the interior effectively damps the motion. The spin-down phenomenon also follows readily from Kelvin's circulation theorem. In the interior of the fluid, where the effect of friction can be neglected, we can apply (1.1.26) and (1.1.27), i.e. G v r k abs ( vabs ) ( f ) const. (2.4.10) where is the area within the closed material curve in the horizontal plane. The vertical flux from the bottom Ekman layer must lead to horizontal motion in the interior; see the sketch in Fig. 2.2. 65 Fig. 2.2. A material, closed curve at consecutive times t1 and t2. In the present example, with a cyclone ( G 0) in the interior, leading to convergence in the Ekman layer, a closed material curve in the interior horizontal plane must be extended, as depicted in Fig. 2.2. Accordingly, the area in (2.4.10) becomes enlarged, and therefore the geostrophic vorticity must decrease, which means spin-down of the cyclone. We realize immediately that if we have an anti-cyclone ( G 0) in the interior, this will result in a divergent mass transport in the bottom layer, and hence contraction of material curves in the interior. According to (2.4.10), the absolute vorticity then must increase. This increase is achieved by the (negative) G approaching zero, i.e. a spin-down of the anti-cyclone. For a variable wind stress, a balance in the interior can be achieved if wB = wS. From (2.4.2) and (2.4.3) we then obtain G 2 k ( S ) . f B (2.4.11) From our previous example with S /(L) ~ 106 cm s2, we find that this balance results in G ~ 4.5105s1. 2.5 Variable vertical eddy viscosity Because of mixing due to the action of wind, breaking of waves etc., the level of turbulence close to the surface is higher than in the interior of the ocean. This means that the vertical eddy viscosity is not a constant, but depends on depth. Let us make a simple study of how the effect of a vertical variation of A influences the Ekman flow in the surface layer in a homogeneous ocean. For that purpose we consider a linear depth dependence: 66 A A0 z , (2.5.1) where can be positive or negative. In this way we model an A that is decreasing or increasing with depth, respectively. We rescale the depth by ~ z z / , 0 (2.5.2) where 0 (2 A0 / f )1 / 2 . Accordingly we can write A A0 (1 ~ z) . (2.5.3) Here is a dimensionless parameter given by 2 A0 f 1/ 2 . (2.5.4) From (2.2.2), with p independent of x and y, we find that (1 ~ z )W~z ~z W~z 2 2iW 0 , (2.5.5) where W = u + iv. We assume a constant wind stress in the y-direction. The boundary conditions then become W~z i T , ~ z 0, (2.5.6) where T S( y ) 0 /( A0 ) . In the deep ocean we assume that W 0, ~ z . (2.5.7) We will assume that is a small parameter (|| << 1), allowing us to expand the solution in a power series after . Accordingly W W ( 0) W (1) 2W ( 2) (2.5.8) We now insert this series into the governing equation (2.5.5) and the boundary conditions (2.5.6) and (2.5.7). Then we collect terms with the same power of . Since this is an arbitrary small parameter, our equations can only be fulfilled if the coefficients multiplying each power of vanish identically. The terms multiplying 0 yield, when set to zero: W~z(~z0 ) 2 2iW ( 0 ) 0, ~ W~( 0 ) iT , z 0, z (0) W 0, ~ z . (2.5.9) This is the ordinary Ekman problem for constant A, and the solution can be written W ( 0) C0 exp m~ z, where m 2 2 2i and C0 iT / m . (2.5.10) 67 Next, the terms multiplying 1 yield, when set to zero: W~z(1~z) 2 2iW (1) zW~z(~z0 ) W~z( 0 ) , ~ W~z(1) 0, z 0, ~ W (1) 0, z . (2.5.11) Here W(0) is given by (2.5.10). It is easy to show that a particular solution can be written Wp(1) D~ z exp m~ z E~ z 2 exp m~ z. (2.5.12) By insertion into (2.5.11) and comparing terms with equal powers of ~z , we find that D C0 / 4, E mC0 / 4 . (2.5.13) A complete solution of (2.5.11) that satisfies W (1) 0, ~ z , can then be written W (1) C1 exp m~ z Wp(1) . (2.5.14) The application of the boundary condition W~z(1) 0, ~ z 0 , finally determines the constant C1. The solution can be written W (1) V0 (1 m~ z m2 ~ z 2 ) exp m~ z, 4 2 (2.5.15) where V0 S( y ) ( A0 f )1 / 2 . (2.5.16) In particular, let us study the surface current, i.e. we take ~ z 0 . The complex surface flow W(0) can be written in the present approximation as W (0) W ( 0) (0) W (1) (0) V0 1 i . 4 2 (2.5.17) From the real and imaginary parts we obtain V0 1 , 2 4 V v(0) 0 . 2 u (0) (2.5.18) If we had continued our approximations, the next terms in (2.5.18) would have been proportional to 2. If is sufficiently small, these terms can be neglected. The surface current is deflected an angle 0 to the right of the wind. This angle is given by 0 arctan u (0) arctan 1 v(0) 4 . (2.5.19) 68 We then observe that if < 0 (A increases with depth), we have 0 < 45. For > 0 (A decreases with depth), we have 0 > 45. We must remember that this result has been derived under the assumption that || << 1, or 1/ 2 A f 0 2 2 . (2.5.20) Observations close to the surface often show a tendency for wind-driven ocean currents (when geostrophic components are removed) to be more in the direction of the wind than what follows from an Ekman analyses with constant eddy viscosity. This can partly be due to the fact that some of the transport is caused by surface wind waves in a direction close to the wind. However, it also suggests that the turbulent eddy viscosity must increase with depth in the upper part of the surface layer, as indicated by the present analysis. Since the eddy coefficients are small in the interior of the ocean, the eddy viscosity must therefore increase to a maximum value at a certain depth, before it decreases to attain its deep ocean value. 2.6 Equations for the volume transport At the free surface z ( x, y, t ) , we neglect any turbulent fluctuations. This means that can be associated with the mean value defined by (2.1.1). By integrating the continuity equation v 0 for the mean motion in the vertical, and applying the mean versions of the boundary conditions (1.1.6) and (1.1.8), we find t u dz v dz . x H y H (2.6.1) This is identical to (1.1.10), but the dependent variables here are mean quantities. When we take into account the effects of friction and a variable density, u and v are no longer independent of z, as in section 1. We now introduce volume transports per unit length. In the x- and y-direction they are, respectively H V v dz. H U u dz, (2.6.2) Accordingly, we can write (2.6.2) as t U x V y . (2.6.3) 69 In the momentum equation we apply the Boussinesq approximation, i.e. we assume that the density changes are only important in connection with the action of gravity. This means that in (2.1.11) we can use = r, where r is a constant reference density. Integrating the acceleration term in (2.1.11), we find 2 H(ut v u)dz U t x Hu dz y Huv dz . (2.6.4) From the assumption of hydrostatic balance in the vertical, we can write the pressure as p PS ( x, y, t ) g dz . (2.6.5) z Let us now consider the x-component of (2.1.11). By integrating the pressure term vertically, and applying the Boussinesq approximation, we find that 1 p H x dz 1 r p x dz H 1 r I x( p ) 1 r PS x 1 r PB H x . (2.6.6) Here PB and I(p) are the pressure at the bottom and the depth-integrated pressure, respectively. They are defined by PB PS g dz, H ( p) I p dz. H (2.6.7) The friction term containing the vertical variation becomes ( Au ) z z dz H 1 r ( S( x ) B( x ) ) , (2.6.8) where S( x ) ( x ) ( ) is the wind stress and B( x ) ( x ) ( H ) is the bottom stress in the x-direction. The integrated friction terms containing the horizontal variation are more difficult to write in a simple form. For example, 2 A u dz A u dz u ( ) 2 u ( ) H u ( H ) 2 H u ( H ) , (2.6.9) H xx H xx x x xx x x 2 H x H where we have taken AH to be constant. We will follow Munk (1950), and assume that the first term dominates on the right hand side, i.e. A H H u xx dz AH U xx . (2.6.10) 70 Quite similar results are obtained for the y-component of (2.1.11). Furthermore, we assume that the integrated advection terms in (2.6.4) are small compared to the integrated Coriolis term (i.e., we assume small Rossby numbers). The integrated momentum equation can then be written r (U t fV ) I x( p ) PS x PB H x S( x ) B( x ) r AH 2HU , r (Vt fU ) I y( p ) PS y PB H y S( y ) B( y ) r AH 2HV . (2.6.11) These equations are very useful for describing oceanic transports. Even though we have applied the Boussinesq approximation, they are valid for an ocean with variable density, thus including baroclinic motion; see section 3. In literature one often find these equations written in terms of mass transports (M(x), M(y)) instead of volume transports, as here. The transition is very simple, since M ( x) v dz r v dz r V . H H M u dz r u dz rU , H ( y) H (2.6.12) An interesting challenge of great practical importance is to determine how the surface elevation varies in time and space, when we know the atmospheric surface pressure and the wind stresses from meteorological prognoses. For this particular problem, it turns out that we to a good approximation can neglect the entire density variation, i.e. treat the ocean as a constant density fluid. In that case we obtain from (2.6.5): p PS ( x, y, t ) g r ( z ) . (2.6.13) Furthermore, we assume a simple, linear relation between the bottom stress and the volume flux: B( x ) / r rU / H , B( y ) / r rV / H , (2.6.14) where r is a constant friction coefficient. This is analogous to (2.2.6) for the Ekman current at the bottom, except that here we use the mean velocities U/H and V/H, and not the bottom velocity. By assuming that H , which always is a good approximation, (2.6.11) reduces to 71 r U, r r H H 1 ( y) r Vt fU gH y PSy S V . r r H U t fV gH x H PSx 1 S( x ) (2.6.15) Here we have disregarded horizontal diffusion of momentum, i.e. taken AH = 0. These equations, together with (2.6.3), govern the so-called storm surge problem for an ocean of variable depth. The numerical solution of this system of equations, within the appropriate geometrical domain, is performed routinely on a daily basis in many countries bordering the sea to predict storm-induced water level changes along the coasts. We will not go into the general solution of this problem, but we mention that for a storm area of limited extent, the solution will consist of a combination of free waves “leaking” out of the area (propagating much faster than the storm centre) and a forced solution like Ekman's elementary current system. This part of the solution has the same lateral dimension as the storm area, and it follows approximately the motion of the storm centre. Equations (2.6.15) also describe how the motion becomes attenuated if the atmospheric sea surface pressure becomes constant (PS = 0), and the wind ceases to blow ( S 0 ). Let us assume that H is constant. By introducing the mean vorticity 1 (Vx U y ) , H (2.6.16) we find on an f-plane that t r . H (2.6.17) Accordingly, the integrated motion decreases exponentially with time. This is analogous to what we found in (2.4.7) for the spin-down of a barotropic vortex due to Ekman suction. We note that (2.6.17) is in fact correct even if PS 0, as long as the wind stress has zero vorticity, i.e. if k ( S ) 0 . As a second example of a transient solution of (2.6.15) for constant depth, we consider planetary motion ( 0). We now neglect the effect of bottom friction. For an approximately stationary surface elevation (thereby excluding the presence of Sverdrup waves), (2.6.3) reduces to U x Vy 0 . (2.6.18) 72 We introduce a stream function for the volume transport such that U y , V x. (2.6.19) By obtaining the vorticity from (2.6.15), and inserting from (2.6.19), we find (2.6.20) 2Ht x k ( S / r ) By comparing with (1.9.4), we realize that the homogenous solution of this equation yields Rossby waves. We also note that the vorticity of the wind stress acts as a source term for Rossby waves in the ocean. 2.7 The Sverdrup transport For the rest of section 2 we are going to study ocean circulation. We simplify, and assume that the surface pressure PS and the depth H are constants. Furthermore, we consider stationary motion in the ocean. Then, in terms of the mass transports, (2.6.3) becomes M x( x) M y( y ) 0 , (2.7.1) which naturally leads to the definition of a stream function for the steady mass transport: M ( x ) y , M ( y) x . (2.7.2) For this problem it proves convenient to derive an equation for the vertical vorticity associated with the mass transports. With the present assumptions, and utilizing (2.6.12), we find M ( y ) k ( S ) k ( B ) AH 2H ( M x( y ) M y( x ) ) . (2.7.3) By substituting the stream function from (2.7.2), we obtain a partial differential equation for the single variable : x k ( S ) k ( B ) AH 4H . (2.7.4) Sverdrup (1947) assumed that the effects of friction could be neglected altogether. In that case we need not utilize the stream function, since the mass transport in the northsouth direction M(y) is given directly by (2.7.3), i.e. M ( y ) k ( S ) . (2.7.5) 73 This relation is often referred to as the Sverdrup balance. From (2.7.1) we then obtain that M ( x) x k ( S ) . y (2.7.6) Sverdrup applied these equations to the equatorial region and assumed that the wind (and the wind stress) was purely zonal, i.e. S( y ) 0, S( x ) S( x ) ( y ). (2.7.7) Then (2.7.6) for the east-west transport reduces to M ( x) x 1 d 2 S( x ) . dy 2 (2.7.8) This equation is only 1. order in x, and accordingly we can only apply one boundary condition. We take that M ( x ) 0 at the eastern boundary. If this boundary is located at x = L, the Sverdrup transports become, respectively: M ( x) M ( y) 1 d 2 S( x ) ( x L) . dy 2 1 d S( x ) . dy (2.7.9) (2.7.10) In Fig. 2.3 we have sketched the wind stress distribution in the equatorial region, and the associated mass transport from (2.7.9). Fig. 2.3. Sketch of the wind stress distribution and the mass transport in the equatorial region. 74 Since M(x) 0 for x = 0, this solution is not valid close to the left (western) boundary. We note from (2.7.9) that the equatorial transport is proportional to the 2. derivative (the curvature) of the north-south distribution of the wind stress. By inserting an equatorial easterly wind field (the trade winds) with a local minimum (the doldrums) slightly north of the equator, Sverdrup demonstrated that we in principle obtain the observed equatorial current system with westerly flowing currents north and south of the equator and an easterly flowing counter current sandwiched between them; see the sketch in Fig. 2.3. In reality, the equatorial flow pattern is more complicated, which is probably due to the fact that the real wind system is not as simple as the one used here. An apparent paradox is revealed when we compare the Sverdrup east-west transport, which is along the wind, to the Ekman transport, which is directed 90 to the right of the wind. The solution to this problem lays in the fact that the Sverdrup solution also contains the integrated pressure I(p), defined by (2.6.7). From the stationary version of (2.6.11), with S( y ) = B( y ) = = AH = 0 and H = const., we find I y( p ) f M ( x ) . (2.7.11) f d S( x ) ( x L) C , dy (2.7.12) By inserting from (2.7.9): I ( p) where C is a constant. In this expression, we note that x L. Thus, we see that for a given x, the integrated pressure I(p) has a maximum where d S( x ) / dy has its largest value. This can only be a result of the transport to the right of the wind in the Ekman layer close to the surface; see Fig. 2.4, where we have assumed a simple sinusoidal wind profile. 75 Fig. 2.4. The connection between the wind direction, the Ekman transport and the Sverdrup transport. Accordingly, the Sverdrup transport in the east-west direction results from geostrophic balance in the north-south direction. Finally, we remark that the Sverdrup balance (2.7.5) expresses the fact that the advection of planetary vorticity is equal to the vorticity of the surface wind stress. For an anti-cyclonic wind field, as depicted in Fig. 2.4, we have that k ( S ) = d S( x ) / dy 0 . Hence, a fluid column must move in such a way that it has a southward-directed velocity component, as indicated in the figure. 2.8 Theories of Stommel and Munk (western intensification) As emphasized in the previous section, Sverdrup’s solution for the east-west transport does not satisfy the boundary condition at the western boundary. To remedy this fault, the effect of friction must be taken into account. This was done by Stommel (1948), and by Munk (1950). We return to the more general equation (2.7.4). Stommel (1948) solved it with an idealized wind stress distribution, while taking AH = 0. Munk (1950) neglected the bottom friction, i.e. assumed B 0 , and solved the equation using realistic wind data. Munk's solution reproduces most of the features of the large-scale ocean circulation. However, this solution is quite complicated. We will therefore be content with Stommel’s simplified approach, which yields the main features of the circulation 76 pattern. However, we will return to Munk’s model when we discuss the boundary layer at the western coast, using a simplified wind field. Stommel modelled the bottom stress1 by assuming B( x ) K M ( x ) , B( y ) K M ( y ) , (2.8.1) where K is a constant friction coefficient. The ocean is taken to be a rectangular basin of length L and width D, subject to a zonal wind field of the form: S( y ) 0, S( x ) y 0 cos . D (2.8.2) The ocean geometry and the wind stress are sketched in Fig. 2.5. Fig. 2.5. Ocean basin and wind field for the Stommel solution. With the parameterisations (2.8.1) and (2.8.2), and the assumption AH = 0, (2.7.4) reduces to K 2H x 0 D sin y D . (2.8.3) The boundaries of the model are impermeable, which means that the flow must be purely tangential at any part of the boundary. From the definition of the stream function, this means that the boundary is a streamline given by = 0. We assume that the solution of (2.8.3) can be written 1 Stommel imagined that the lower limit of the vertical integration was taken to be the depth where the current velocity was practically zero. This depth is usually much less than the total depth. In this case, ”the bottom stress” refers to the frictional stress between the dynamically active upper layer, and the passive lower layer. 77 ( x ) sin y D . (2.8.4) This solution satisfies the boundary conditions for y =0, D, i.e. (0) = (D) = 0 for all x. Inserting (2.8.4) into (2.8.3), we find K 2 D 2 0 KD , (2.8.5) with boundary conditions 0, for x 0, L . This equation has a particular solution p 0D , K (2.8.6) while the homogeneous solution can be written h C1 exp( r1 x) C2 exp( r2 x) . (2.8.7) Here 2 2 r1 , r2 2K 4K 2 D 2 1/ 2 . (2.8.8) The constants C1 and C2 are obtained from the boundary conditions. By insertion: h (0) p 0, h ( L) p 0. (2.8.9) 0D {exp( r2 L) 1} , K {exp( r1L) exp( r2 L)} D {exp( r1L) 1} C2 0 . K {exp( r1L) exp( r2 L)} (2.8.10) This yields C1 The complete solution can then be written exp{ ( L x )} sinh( x ) exp{ x}sinh( ( L x)) 0D y 2K 2K sin 1 , (2.8.11) K D sinh( L) where 2 2 2 2 D 4K 1/ 2 . (2.8.12) Let us first see what this solution yields on an f-plane, i.e. when = 0. It is then convenient to place the origin in the centre of the basin. This is done by defining new coordinates: x* x L / 2, y* y D / 2 . (2.8.13) 78 By insertion into (2.8.11), with = 0, we find that L L sinh{ x* } sinh{ x* } D y D 2 D 2 . ( x* , y* ) 0 cos * 1 K D sinh{ L} D (2.8.14) We realize straightaway that the streamlines from (2.8.14) are symmetric about the basin centre. This situation is depicted in Fig. 2.6. Fig. 2.6. Stream lines when the beta-effect is zero. This is not in accordance with our knowledge of the large-scale ocean circulation, which tells us that the currents are much more intense on the western side (the Gulf Stream, Kuroshio) than on the eastern side. We therefore conclude that the beta-effect must be significant. By inserting relevant parameters ( ~ 21011 m1s1, K ~ 106s1, D ~ 6103 km, L ~ 104 km) we find a flow pattern from (2.7.23) as sketched in Fig. 2.7. Fig. 2.7. The flow pattern when the beta-effect is included. 79 In this case the solution clearly reproduces the intensified flow on the western side of the ocean. The northward transport on the western side of the ocean occurs mainly within a thin boundary layer, where the thickness is determined by the effect of friction and the beta-effect. For Stommel’s model, the typical boundary-layer scale S is S ~ K / . (2.8.15) The mass transport M b( y ) in the boundary layer is sketched in Fig. 2.8. We will return to this problem at the end of this section. Fig. 2.8. Sketch of the transport at the western boundary in Stommel’s model. Finally, by a simple boundary-layer analysis we discuss the northward transport at the western boundary in the Munk model ( B 0 , AH 0). By assuming that / x / y in the boundary layer, the leading terms in (2.7.3) becomes AH xxxx x 0 . (2.8.16) Since M ( y ) x , we can insert M b( y ) directly into (2.8.16). Furthermore, the boundary-layer scale M for this problem is defined by M ( AH / )1/ 3 . (2.8.17) By introducing a boundary-layer coordinate ~ x x / M , equation (2.8.16) can be written d 3 M b( y ) M b( y ) 0 . 3 ~ dx This equation has a solution of the form M b( y ) C1 exp( r1~ x ) C2 exp( r2 ~ x ) C3 exp( r3 ~ x) , (2.8.18) (2.8.19) 80 where 1 31 / 2 1 31 / 2 . r1 1, r2 i , r3 i 2 2 2 2 (2.8.20) For the solution of (2.8.18) we must require that M b( y ) is finite when ~ x , and that M b( y ) 0 when ~ x 0 (no-slip). Accordingly, C1 = 0 and C2 = C3 in (2.8.19). The solution can then be written as 3 ~ M b( y ) C exp( ~ x / 2) sin x . 2 (2.8.21) Here, the constant C must be determined by matching with the solution valid outside the boundary layer, i.e. the Sverdrup transport. Utilizing the simplified zonal wind stress (2.8.2), (2.7.10) yields ( y) M Sv 1 0 y sin , M x L. D D (2.8.22) In a closed basin, the total north-south transport must be zero. This constitutes the required matching condition, i.e. L 0 M ( y) ( y) ~ M b M dx M Sv dx 0 . (2.8.23) Insertion from (2.8.21) and (2.8.22) allows us to determine C. When we utilize that M << L, the mass transport in the boundary layer becomes M b( y ) 2 0 L x exp 3 M D 2 M 3x y sin sin . 2 M D We have sketched this solution in Fig. 2.9. Fig. 2.9. Sketch of the transport at the western boundary in the Munk model. (2.8.24) 81 The dimensional thickness xb of the layer where the transport is northward, is given by xb 2 3 M . (2.8.25) With AH ~ 103 m2s1 and = 21011m1s1, we find that xb ~ 134 km. This fits well with the width of the Gulf Stream. A similar boundary-layer analysis for the Stommel model yields that M b( y ) 0 L x exp K D S y sin , D (2.8.26) where S = K/. If we choose a typical boundary-layer thickness xb S , our former values for the parameters of the problem gives that xb ~ 160 km. It is worth noticing that in the strong shear in the western boundary current (see Fig. 2.9), the Rossby number R ~ v v /( f k v ) is of the same order of magnitude as the horizontal Ekman number E H ~ AH 2 v /( f k v ) . This means that also the nonlinear terms in the momentum equation may be important in this region. 3. BAROCLINIC MOTION 3.1 Two-layer model We now proceed to study the effect of vertical density stratification in the ocean. In many situations the density is approximately constant in a layer close to the surface, while the density in the deeper water is also are constant (and larger). The transition zone between the two layers is called the pycnocline. Thin pycoclines are typically found in many Norwegian fjords. In extreme cases we can imagine that the pycnocline thickness approaches zero, resulting in a two-layer model with a jump in the density across the interface between the layers. We start out by studying such a model. For simplicity we describe the motion in reference system as shown in Fig. 3.1, where the x-axis is situated at the undisturbed interface between the layers. The constant density in each layer is 1 and 2, respectively, where 2 > 1. 82 Fig. 3.1. Model sketch of the two-layer system. We assume hydrostatic pressure distribution in each layer. By applying that the pressure is PS along the surface, and continuous at the interface, i.e. p1(z = ) = p2(z = ), we find that p1 1 gz 1 g ( H 1 ) PS , p2 2 gz g ( 2 1 ) 1 g ( H 1 ) PS . (3.1.1) We average the motion in the upper and lower layer: (uˆ1 , vˆ1 ) 1 h1 H 1 (u , v ) dz , 1 1 (3.1.2) 1 (uˆ2 , vˆ2 ) (u2 , v2 ) dz , h2 H 2 (3.1.3) Here, h1 H1 and h2 H2 are the total depths of the upper and lower layers, respectively. We assume that our equations can be linearized, i.e. we neglect the convective accelerations. Furthermore, we will disregard the effect of the horizontal eddy viscosity. This means that AH = 0 in each layer. By introducing volume transports (U1 ,V1 ) h1 (uˆ1 , vˆ1 ), (U 2 ,V2 ) h2 (uˆ2 , vˆ2 ), the momentum equation for the upper layer becomes: (3.1.4) 83 1 1 1 h1 1 ( y) 1 ( y) V1t fU1 gh1 y PS y S i . 1 1 1 U1t fV1 gh1 x h1 PS x 1 S( x ) 1 i( x ) , (3.1.5) Here, ( S( x ) , S( y ) ) are the wind stresses along the surface, and ( i( x ) , i( y ) ) are the internal frictional stresses between the layers. Equivalently, for the lower layer we find 1 h 1 1 gh2 x g*h2 x 2 PS x i( x ) B( x ) , 2 2 2 2 1 h2 1 ( y) 1 ( y) V2t fU 2 gh2 y g*h2 y P B , 2 2 S y 2 i 2 U 2t fV2 (3.1.6) where ( B( x ) , B( y ) ) are the frictional stresses at the bottom. Furthermore, we have defined 1 g , g* 2 2 (3.1.7) which is referred to as the reduced gravity, because the fraction ( 2 1 ) / 1 is small for typical ocean conditions. By integrating the continuity equation u x v y wz 0 in each layer, we find, without any linearization of the boundary conditions, that t t U1x V1 y , t U 2 x V2 y . (3.1.8) Barotropic response Assume that the mean velocities in each layer are approximately equal, i.e. uˆ1 uˆ2 , vˆ1 vˆ2 . This leads to U1 U 2 , H1 H 2 V1 V 2 , H1 H 2 (3.1.9) when we assume that , H1, H 2 . For simplicity, we also take that the lower layer has a constant depth. From (3.1.8) we then obtain t t or H1 H (U 2 x V2 y ) 1 t , H2 H2 (3.1.10) 84 H2 . H1 H 2 (3.1.11) Here the integration constant must be zero when we consider wave motion. We note from (3.1.11) that and are in phase, and that | | < | |. By neglecting the effect of the earth’s rotation, assuming constant surface pressure and neglecting frictional effects in (3.1.5), equations (3.1.9) and (3.1.11) yield tt g ( H1 H 2 ) xx 0 , (3.1.12) when / y 0 . The solution is F1 ( x c0t ) F2 ( x c0t ) , (3.1.13) where c02 g ( H1 H 2 ) . The expression (3.1.13) describes long surface waves propagating in a non-rotating canal with depth H1 + H2. This is the solution we would have found if we, as a starting point, had neglected the density difference between the layers; see the one-layer model in section 1. Such a solution (a free wave), which is not influenced by the small density difference between the layers, is often referred to as the barotropic response. The original meaning of the word “barotropic” is related to the field of mass, and expresses the fact that the pressure is constant along the density surfaces, i.e. the isobaric and isopycnal surfaces coincide. Mathematically this can be expressed as p 0 . This was the case for the free waves in section 1, where the density was constant everywhere, and the pressure was constant along the sea surface. For the two-layer model this would mean that the pressure should be constant along the interface between the two layers. This is only approximately satisfied here, but nonetheless it has become customary to denote the response in this case as the barotropic response. Baroclinic response We now assume that t t . Then, from (3.1.8): (U 1 U 2 ) x (V1 V2 ) y 0 . (3.1.14) For simplicity we take the bottom to be flat. A particular solution of (3.1.14) can be written 85 U1 U 2 , V1 V2 , (3.1.15) i.e. the volume fluxes are equal, but oppositely directed in each layer. By taking the surface pressure to be constant, and neglecting the effect friction, summation of (3.1.5) and (3.1.6) yields H H 2 1 1 2 , ( 2 1 ) H 2 (3.1.16) where, as in the barotropic case, the integration constant must be zero. Furthermore, we have used that h1 H1, h2 H2. The difference between 1 and 2 is quite small, which allows us to use the approximations 2 1 = , and 1 ~ 2 . Thus, equation (3.1.16) can be rewritten as H1 H 2 . H 2 (3.1.17) We note that and are oppositely directed, and that | | >> | |, as initially assumed. Assuming that / y 0 , we obtain from (3.1.6), (3.1.8), and (3.1.17) that tt c12 xx 0 , (3.1.18) where c12 g * H1 H 2 . H1 H 2 (3.1.19) The solution of (3.1.18) can be written F1 ( x c1t ) F2 ( x c1t ) . (3.1.20) This represents internal gravity waves propagating with phase speed c1 along the interface between the layers; see the sketch in Fig. 3.2. Fig. 3.2. Internal wave in a two-layer model. 86 The solution to (3.1.20) is often called the baroclinic response. As for barotropic, the term “baroclinic” is linked to the mass field. In a baroclinic mass field the constant pressure surfaces and the constant density surfaces intersect, i.e. p 0 . Equation (3.1.17) shows that this is the case here, since when > 0, then < 0. Accordingly, the pressure varies along the interface, which is a constant density surface. Let us assume that the lower layer is very deep, i.e. H2 >> H1. This is the most common configuration in the ocean. From (3.1.6) we find for the x-component in the lower layer 1 1 11 1 g x g* x PS x i( x ) B( x ) U 2t f V2 . 2 2 h2 2 2 (3.1.21) For the baroclinic case, U2 and V2 are finite when h2 , and so are the frictional stresses. Accordingly, for this limit, (3.1.21) reduces to g x 2 1 g* x PS x . 1 1 (3.1.22) In the same way we find for the y-component g y 2 1 g* y PS y . 1 1 (3.1.23) By inserting (3.1.22) and (3.1.23) into (3.1.5) for the upper layer, we find that 2 1 1 h1 g* x S( x ) i( x ) , 1 1 1 1 1 V1t f U1 2 h1 g* y S( y ) i( y ) . 1 1 1 U1t f V1 (3.1.24) For the baroclinic case, the depth of the upper layer can be written h1 H1 H1 . Furthermore, we apply that 2 1 , and 1 2 = . By linearizing the pressure term (the first term on the right-hand side), equations (3.1.24) and (3.1.8) yield U1t f V1 g* H1h1x 1 S( x ) i( x ) , 1 V1t f U1 g* H1h1 y 1 1 h1t U1x V1 y . ( y ) i( y ) , S (3.1.25) These equations for the baroclinic response in the upper layer are formally identical to the equations describing the storm surge problem for a homogeneous ocean; see 87 (2.6.15), when the upper layer thickness replaces the surface elevation, and the gravity g is replaced by g * . The set of equations (3.1.25) describes what is often referred to as a reduced gravity model for the volume transport in the upper layer. Even though the numerical values for the volume fluxes in the lower layer are of the same order of magnitude as in the upper layer, the mean velocity in the lower layer is negligible, since H2 . Therefore, we usually say that the lower layer has no motion in this approximation. We immediately realize from (3.1.25) that transient phenomena such as Sverdrup-, Kelvin- and planetary Rossby waves in a rotating ocean of constant density have their internal (baroclinic) counter-parts in a two-layer model. The analysis for the internal response is identical to the analysis is section 1. It often suffices to replace g with g * and H with H1 in the solution for the barotropic response. Analogous to the barotropic case we can define a length scale a1 that characterizes the significance of earth’s rotation. We write a1 c1 / f , (3.1.26) where c12 g * H 1 . The length scale a1 is called the internal, or baroclinic, Rossby radius. Typical values for c1 in the ocean are 2-3 m s1. Hence, a1 20-30 km, which is much less than the typical barotropic Rossby radius. Therefore, the effect of earth’s rotation will be much more important for the baroclinic response than for the barotropic one with the same horisontal scale, or wavelength. 3.2 Continuously stratified fluid We now turn to the more general problem of a continuous density stratification, and start by investigating the stability of a stratified incompressible fluid under the influence of gravity. The equilibrium values are: v 0, 0 ( z ), p p0 ( z ) g 0 dz const . (3.2.1) We introduce small perturbations (denoted by primes) from the state of equilibrium, writing the velocity, density, and pressure as 88 v v ( x, y, z, t ), 0 ( z ) ( x, y, z, t ), p p0 ( z ) p( x, y, z, t ). (3.2.2) We assume that the density is conserved for a fluid particle. Furthermore, we take that the perturbations are so small that we can linearize our problem, i.e. neglect terms that are products of perturbation quantities. The equations for the conservation of momentum, density, and mass then reduce to 0 ( z )(u t fv) p x z( x ) , (3.2.3) 0 ( z)(vt fu) p y z( y ) , (3.2.4) 0 ( z )wt p z g , t d0 w0, dz u x v y wz 0 . (3.2.5) (3.2.6) (3.2.7) Here we have for simplicity left out the primes that mark the perturbations. In (3.2.3) and (3.2.4) ( x ) and ( y ) represent the tangential stresses in the fluid. We have also assumed that the vertical variations of these stresses are much larger than the corresponding horisontal variations. Furthermore, we have neglected the effect of friction in vertical component of the momentum equation (3.2.5). 3.3 Free internal waves in a rotating ocean We start by disregarding completely the effect of friction on the fluid motion, i.e. we take ( x ) ( y ) 0 in (3.2.3) and (3.2.4). Furthermore, we introduce the BruntVäisälä frequency (or the buoyancy frequency) N, defined by N 2 ( z) g d 0 . 0 dz (3.3.1) We are here going to study motion in a stably stratified incompressible fluid. In this case we must have that d0 / dz 0 , meaning that N is real and positive. Equation (3.2.6) can then be written g t N 2 0 w 0 . (3.3.2) By differentiating (3.2.5) with respect to time, and utilizing (3.3.2), we find that 0 (wt t N 2 w) p z t . (3.3.3) 89 From this equation we note that the time scale for pure vertical motion ( pzt 0 ) is N 1 . Elimination of the gradient from (3.2.3)-(3.2.4), yields the vorticity equation. On an f-plane we obtain (vx u y )t fwz , (3.3.4) where we have applied (3.2.7). Forming the horizontal divergence from the same two equations, and applying (3.2.7), we find 0 ( z){wzt f (vx u y )} 2H p . (3.3.5) Elimination of the vorticity from the equations above, yields 0 wz t t f 2 wz 2H pt . (3.3.6) Finally, by eliminating the pressure between (3.3.3) and (3.3.6), we obtain 2 1 2 2 2 1 ( w ) 0 . H w ( 0 wz ) z N H w f 0 0 0 z z t t (3.3.7) We simplify (3.3.7) by assuming that 0(z) varies slowly over the typical vertical scale for w, i.e. 1 0 ( 0 wz ) z wz z . (3.3.8) This is Boussinesq approximation for internal waves. By introducing the BruntVäisälä frequency (3.3.1), we can write N2 ( 0 wz ) z wz wz z . 0 g 1 (3.3.9) We realize from (3.3.8) that the Boussinesq approximation implies that N2 wz wz z . g (3.3.10) If d is a typical vertical scale for the motion, the above equation yields N 2 g / d , (3.3.11) where d max H . For a shallow ocean we typically have that g/H ~ 101s2, while for a deep ocean (H = 4000 m), the corresponding value becomes g / H ~ 14 102s2. Measurements in the ocean show that N 2 ~ 104 106s2, so (3.3.11) is usually very well satisfied. We will therefore utilize the Boussinesq approximation in the future analysis of this problem. Equation (3.3.7) then reduces to 2 wt t N 2 ( z) 2H w f 2 wz z 0 , (3.3.12) 90 where 2 is the three-dimensional Laplacian operator. We derive the same equation by letting 0(z) r on the left-hand sides of (3.2.3)-(3.2.5), where r is a constant reference density. Then N 2 ( g / r )d0 / dz . This latter approach is probably the most common one when applying the Boussinesq approximation. We assume that the ocean is unlimited in the horizontal direction, and consider a wave solution of the form w W ( z ) exp( i (kx t )) . (3.3.13) Here the x-axis is directed along the wave propagation direction. From (3.3.12) we then obtain N 2 2 W k 2 2 W 0, 2 f (3.3.14) where a prime denote differentiation with respect to z. Constant Brunt-Väisälä frequency Later on we shall allow N to vary with z. In this treatment, we simplify, and assume that N is constant. Typical values for N and f in the ocean (and atmosphere) are N ~ 102s1 and f ~ 104s1, i.e. N >> f. From equation (3.3.14) we then note that we have wave solutions in the z-direction if f < < N, while for < f or > N, the solutions must be of exponential character in the z-direction. Let us assume that f < < N. We then take W exp( im z ) , (3.3.15) where m is a wave number in the vertical direction. By insertion into (3.3.14), we obtain the dispersion relation 2 k 2 N 2 m2 f 2 . k 2 m2 (3.3.16) We realize that we have an-isotropic system, since cannot be expressed solely as a function of the magnitude of the wave number vector. Analogous to the discussion of planetary Rossby waves in section 1.9, we can define a wave number vector as ( k , m) . Then the phase speed and group velocity, become, respectively c / 2 , (3.3.17) (3.3.18) 91 and c g . (3.3.19) From the last relation we notice that the group velocity is always directed towards increasing values of . When is constant, we find from (3.3.16) that this gives isolines that are straight lines in the wave number space; see also the sketch in Fig. 3.3. Fig. 3.3. Lines of constant frequency for internal waves with rotation in the twodimensional wave number space. Along the m-axis (where k = 0), we have = f (small). Along the k-axis (where m = 0), we have = N (large). Hence, the direction of the group velocity becomes as shown in the figure, while the phase speed from (3.3.18) is directed along the wave number vector. If we imagine that the wave number m is given, we can plot as a function of k, as depicted in Fig. 3.4. Fig. 3.4. Dispersion diagram for internal waves with rotation. 92 We can define a Rossby radius of deformation for internal motion with vertical wave number m by ai N . mf (3.3.20) For k ai1 , the effect of rotation dominates (compare with Fig. 1.6 for the barotropic case). In the ocean, the wave number m cannot be chosen arbitrarily, since the vertical distance is limited by the depth. If we, for simplicity, disregard the surface elevation and assume a constant depth, we must have that w = 0 for z = 0, H; see Fig. 3.5. Fig. 3.5. Internal waves in an ocean with horizontal surface and horizontal bottom. A solution of (3.3.14), which satisfies the upper boundary condition, is W C sin m z , (3.3.21) where N 2 2 . m k 2 2 f 2 2 (3.3.22) For the solution to satisfy the boundary condition at z H , we must require m n , H n 1, 2, ... (3.3.23) This means that vertical wave number must form a discrete (but infinite) set. Equation (3.3.22) then yields for the frequency k 2 N 2 f 2 n 2 2 / H 2 2 2 2 2 k n /H 1/ 2 . (3.3.24) 93 We see that, for a disturbance with a given wave number in the horizontal direction, the system (ocean) responds with a discrete number of eigenfrequencies. The solution for w in this case can be written w C sin( m z ) exp{i (kx t )} C exp{i (kx mz t )} exp{i(kx mz t )} 2i (3.3.25) The latter expression can be interpreted as the superposition of waves in a horizontal layer consisting of an incoming, obliquely upward propagating wave, and an obliquely downward reflected wave, where m must attain the value (3.3.23) for the wave system to satisfy the boundary condition at the bottom. We now consider the case where the motion is mainly horizontal. This allows us to disregard the vertical acceleration in the momentum equation, i.e. we apply the hydrostatic approximation. Accordingly, in (3.3.3) we take that wt t N 2 w , (3.3.26) which leads to N 2w 1 0 pz t . (3.3.27) From (3.3.26) we realize that the hydrostatic approximation leads to the condition 2 N 2 . (3.3.28) From Fig. 3.4 we note that this implies that k << m, i.e. the horizontal scale of motion is much larger than the vertical scale. Since the depth H yields the upper limit for the vertical scale, disturbances with wavelength >> H will satisfy the hydrostatic condition. This requirement applies to barotropic surface waves as well as baroclinic internal waves. Applying the hydrostatic approximation, (3.3.16) reduces to k2N 2 f 2 m2 1/ 2 . (3.3.29) This is the frequency for internal Sverdrup waves. For an ocean with depth H and a horizontal surface, i.e. m = n/H as in equation (3.3.23), we can write ( f 2 c n2 k 2 )1 / 2 . (3.3.30) Here cn HN /( n ), n 1, 2, 3, ... (3.3.31) which is the phase speed for long internal waves in the non-rotating case. Since 94 m n / H N / cn , (3.3.32) equation (3.3.20) yields the internal (baroclinic) Rossby radius ai an cn / f , n 1, 2, 3, ... (3.3.33) We note that this is analogous to the definition of the barotropic Rossby radius appearing in (1.3.14). For one single internal mode, i.e. a two-layer structure, this is similar to (3.1.26). 3.4 Internal response to wind forcing; upwelling at a straight coast We apply the set of equations (3.2.3)-(3.2.7), and utilize the Boussinesq approximation and the hydrostatic approximation, i.e. ut fv vt fu 1 S 1 S 1 px z( x ) , S py 1 S ( y) z , p z g . (3.4.1) (3.4.2) Here the constant surface density S is used as a reference density. Furthermore, we introduce the vertical displacement ( x, y, z , t ) of a material surface, so that w = D/dt in the fluid. Linearly, this becomes w t . (3.4.3) The conservation of density (3.2.6) then yields for the density perturbation S g N 2 . (3.4.4) where we have assumed that = = 0 at t = 0. Inserting into (3.4.2), we obtain p z S N 2 , (3.4.5) while the continuity equation can be written z t u x v y . (3.4.6) In general, we take that N = N(z), and we write the solutions to our problem as infinite series. For simplicity, we assume that the depth is constant, and that the surface is horisontal at all times. Accordingly: 0, z H , 0. (3.4.7) In principle, it is also possible to allow the surface to vary in time and space. However, the solution shows that the internal response can be acchieved, to a good 95 approximation, by assuming a horisontal surface (the rigid lid approximation); see Gill and Clark (1975). According to our adopted approach, we write the solutions as n 1 v vn ( x, y, t ) n ( z ), n 1 pn S pn ( x, y, t ) n ( z ), n1 n ( x, y, t ) n ( z ), n 1 u un ( x, y, t ) n ( z ), (3.4.8) where the primes denote differentiation with respect to z. By inserting the solutions into (3.4.5), we find n p ( x, y, t ) ( z) p n 1 n n n N 2 ( z ) 0 . (3.4.9) For the variables to separate, we must require n pn const. 1 . cn2 (3.4.10) Furthermore, for (3.4.9) to be satisfied for all x, y and t, we must have that n N2 n 0 . cn2 (3.4.11) The boundary conditions (3.4.7) yield n 0, z H , 0 . (3.4.12) Equation (3.4.11) and the boundary conditions (3.4.12) define an eigenvalue problem, i.e. for given N = N(z) we can, in principle, determine the constant eigenvalues cn, and the eigenfunctions n (z ) , which appear in the series (3.4.8). It is easy to demonstrate that the differentiated eigenfunctions n constitute an orthogonal set. Since (3.4.11) is valid for arbitrary numbers n and m, we can write cn2n N 2n 0, cm2 m N 2m 0, (3.4.13) where m n. We multiply the upper and lower equations by m and n, respectively. By subtracting and integrating from z = H to z = 0, utilizing (3.4.12), we find 0 (c c ) nm dz 0 . 2 n 2 m H Accordingly, for n m, i.e. cn cm, we must have that (3.4.14) 96 0 dz 0, n m, n m (3.4.15) H which proves the orthogonality. Since the eigenfunctions are known, apart from multiplying constants (as for all homogeneous problems), we can normalize them by assuming, for example, that 0 2 n dz H H . 2 (3.4.16) This procedure is generally valid for N = N(z). To exemplify, and discuss explicit solutions in a simple way, we assume that N is constant. Then the eigenfunctions become N cn n An sin z , (3.4.17) which satisfies equation (3.4.11) and the upper boundary condition. The requirement n(H) = 0 yields the eigenvalues: HN n , cn (3.4.18) or cn HN /( n ) , which is identical to (3.3.31). Finally, the normalization condition (3.4.15) gives An = H/(n). We now insert the series (3.4.8) into (3.4.1) and (3.4.6), and multiply each equation with 1 , 2 , 3 , etc. By integrating from z = H to z = 0, and applying the orthogonality condition (3.4.15), we finally obtain un fvn cn2 n n( x ) , t x vn 2 n fun cn n( y ) , t y n un vn , t x y (3.4.19) where ( x) n ( y) n 2 S H 2 S H ( x ) dz , n z H 0 ( y) H z n dz. 0 (3.4.20) We notice from (3.4.19) that this set of equations is formally identical to the equations for the barotropic volume transports driven by surface winds, e.g. (2.6.15). 97 The problem in the baroclinic case is related to the form of the horizontal shear stresses ( x ) and ( y ) in the fluid. In principle, these are unknown, and depending on the fluid motion. However, we shall simplify the problem by assuming that we know the vertical gradients of these stresses in the fluid. Assume that a constant wind is blowing along a straight coast, so that the surface wind stresses become S( x ) 0, S( y ) = 0; see the sketch in Fig. 3.6. The model is situated in the northern hemisphere, i.e. f > 0. Fig. 3.6. Model sketch of upwelling/downwelling at a straight coast. We assume that the shear stresses are only felt in a relatively thin layer close to the surface, i.e. the mixed layer, with a thickness d << H. Here the stresses vary linearly with depth: ( x) z d , S d 0, ( y) 0, ( x) d z 0, H z d , H z 0. (3.4.21) With this variation in z, (3.4.20) yields 2 S( x ) n (d ), S Hd 0. n( x ) ( y) n (3.4.22) We assume that the solutions are independent of the along-shore coordinate x, i.e., from (3.4.19): 98 un fvn n( x ) , t vn fun cn2 n , t y n vn . t y (3.4.23) These equations have a particular solution where vn is independent of time. By assuming that vn / t 0 , and eliminating un and n from the equations above, we find 2 vn 1 f 2 v n 2 n( x ) , 2 y an cn (3.4.24) where an cn / f is the Rossby radius for internal waves. By requiring that vn 0, y 0, vn finite, y , (3.4.25) the solution of (3.4.25) becomes vn n( x ) f 1 exp( y / an ) . (3.4.26) From (3.4.23) we then obtain un n( x )t exp( y / an ), ( x )t n n exp( y / an ). f an (3.4.27) Thus, un and n increase linearly in time during the action of the wind. From the derived solution we see that a wind parallel to the coast results in upwelling or downwelling within an area limited by the coast and the baroclinic Rossby radius. Within this area we also notice the presence of a jet-like flow un parallel to the coast. This flow is geostrophically balanced; see the second equation in (3.4.23) with vn / t 0 . We now discuss our solution in some more details. For this purpose the first term in the series (3.4.8) for v and suffice: v v11( z ) ... 11 ( z ) ... (3.4.28) To simplify, we again take that N is constant. Then, from (3.4.17), (3.4.18) and (3.4.22): 99 z sin , H c1 HN / , ( x) 2 d 1( x ) S sin . S d H 1 H (3.4.29) By inserting into (3.4.28), we find 2 S( x ) d z sin 1 exp( y / a1 ) cos ... S f d H H ( x) 2 S d z w t sin exp( y / a1 ) sin ..... S N d H H v (3.4.30) Here –H z 0 and f > 0. For wind in the negative x-direction ( S( x ) < 0), we find that w 0 in the region limited by baroclinic Rossby radius. Accordingly, the Ekman surface layer transport away from the coast leads to a compensating flow from below (upwelling). This is consistent with the sign of v in (3.4.30), since v is positive close to the surface and negative near the bottom; see the sketch in Fig. 3.7. Fig.3.7. Sketch of an upwelling situation. We finally mention that since the x-component u and the vertical displacement increase linearly in time, the theory developed here is only valid as long as the nonlinear terms in the equations remain small. We return for a moment to the two-layer reduced gravity model to find out what this would yield under similar conditions. By assuming i( x ) = i( y ) = S( y ) = V1t = 0 in (3.1.26), we find, analogous to (3.4.24): V1 yy 1 f S( x ) V , 1 a12 c12 (3.4.31) 100 when we take that / x 0 . The solution becomes V1 S( x ) 1 exp( y / a1 ), f S( x ) t exp( y / a1 ), ( x) h1 S t exp( y / a1 ), c1 U1 (3.4.32) where c1 = ( g * H1)1/2 and a1 = c1/f. We may define an upwelling velocity, when S( x ) < 0, as w1 h1t S( x ) exp( y / a1 ) . c1 (3.4.33) We can now compare with the case of continuous stratification. First, we assume that the layer of frictional influence is thin, i.e. d << H. Furthermore, we insert for z = H/2 to obtain the maximum vertical velocity. From (3.4.19) we then obtain 2 S( x ) w( z H / 2) exp( y / a1 ) . S c1 (3.4.34) Here c1 = HN/ from (3.4.29). By comparing with (3.4.33), we see that the upwelling velocities are remarkably similar, even though (3.4.34) is obtained from the first term in a series expansion. We will not go into further details of this problem. However, it is appropriate to emphasize that this phenomenon is important for marine life. The water that upwells is coming from depths below the mixed layer, and is rich in nutrients. Hence, the upwelling process brings colder, nutrient-rich water to the euphotic zone, where there is sufficient light to support growth and reproduction of plant algae (phytoplankton). This means that upwelling areas are rich in biologic activity. Some of the world’s largest catches of fish are made in such areas, e.g. off the coasts of Peru and Chile. 3.5 Baroclinic instability When studying barotropic instability, we found that, under certain circumstances, small perturbations in a homogeneous fluid could grow in time by extracting energy from the kinetic energy of the basic flow. In a stratified, rotating fluid, where the isobaric and the isopycnal surfaces are not parallel, there is another mechanism for instability. In this case, small perturbations can grow by extracting energy from the potential energy of the basic state. This instability is called baroclinic instability. 101 We consider the simple model described by Eady (1949). The geometry is depicted in Fig. 3.8, and the isopycnals are assumed to be straight lines. Fig. 3.8. Eady’s model. The density 0 in the basic state can be written 0 r (1 y z ) , (3.5.1) where , > 0 and r is a constant reference density. We assume that the basic state is geostrophically balanced, and the vertical pressure distribution is hydrostatic. Utilizing the Boussinesq approximation, the velocity- and pressure distributions can be written g z v 0 Ui i, f p0 p r g r ( z yz 12 z 2 ) , (3.5.2) (3.5.3) respectively, where pr is a constant reference pressure. We introduce small perturbations to this basic state, i.e. we write v v0 ( z ) v ( x, y, z, t ), 0 ( y, z ) ( x, y, z, t ), p p0 ( y, z ) p ( x, y, z, t ). (3.5.4) Again, we assume that the (primed) perturbations are small enough for the equations to be linearized. Furthermore, we disregard the effect of friction. The components of the momentum equation can thus be written, respectively, as ut Uux w dU 1 fv p , dz r x vt Uv x fu 1 r py , (3.5.5) (3.5.6) 102 wt Uwx 1 r pz g. r (3.5.7) Here, for simplicity, we have skipped the primes associated with the perturbations. Furthermore, we assume that the density is conserved for a particle, i.e. D / dt 0 . Linearly, this means that t U x v 0 w 0 0. y z (3.5.8) The continuity equation then becomes u x v y wz 0 . (3.5.9) We assume that f is constant. Equations (3.5.5) and (3.5.6) thus yield g U w y f wz 0 , x f t (3.5.10) where v x u y . Here, we have also applied (3.5.2) and (3.5.9). In studying this problem we assume that the development in time is slow, and that the velocities and the velocity gradients are small. We may then assume that the horizontal velocity perturbations are quasi-geostrophic, i.e. 1 py , f r 1 v px . f r u (3.5.11) Furthermore, we assume that the perturbation pressure is approximately hydrostatic in the vertical: 1 pz . g (3.5.12) When we apply the geostrophic assumption (3.5.11), the vorticity equation (3.5.10) reduces to g 2 U H p f r w y fwz 0 . x f t (3.5.13) As far as the last two terms are concerned, we have that g wy f g H / f U max ~ . fwz fL fL (3.5.14) 103 In applying a quasi-geostrophic approximation, we have implicitly assume that the Rossby number R0 = U/(fL) is much smaller than one. Thus, we can neglect the third term in (3.5.13) compared to the fourth term. This equation then reduces to 2 U H p r f 2 wz 0 . x t (3.5.15) Inserting for v and from (3.5.11) and (3.5.12) into (3.5.8), yields U p z z g r wz 0 , x t (3.5.16) where we also have applied (3.5.1) and (3.5.2). Finally, by eliminating wz between (3.5.15) and (3.5.16), we obtain a single equation for the perturbation pressure: N2 U p z z 2 2H p 0 . x f t (3.5.17) Here we have used that the Brunt-Väisälä frequency N can be written g 0 g . r z N2 (3.5.18) We consider a wave solution in x and t. Then, in general, the differential operator in (3.5.17) yields a non-zero factor. Accordingly, for (3.5.17) to be fulfilled for an arbitrary wave component, we must have that pz z N2 2 H p 0 . f2 (3.5.19) At the boundaries we assume that: v 0, w 0, y 0, L, z 0, H . (3.5.20) These are pure kinematic conditions. For this problem, they must be related to the perturbation pressure at the boundaries, in order to solve (3.5.19). At the sidewalls, a vanishing v in the geostrophic approximation (3.5.11) yields that p 0, y 0, L . (3.5.21) By inserting a vanishing w at the top and bottom boundaries into the density conservation equation (3.5.8), we find: g p x 0, U pz x f t z 0, H . Here we have also utilized (3.5.1), (3.5.11) and (3.5.12). A wave solution that satisfies (3.5.21) can be written (3.5.22) 104 p P( z ) sin( ly ) exp{ik ( x ct )} , (3.5.23) where l n / L for any integer n. We take that k is real, while c can be complex. By inserting this solution into (3.5.19), we find d2 P 4q 2 P 0 , dz 2 (3.5.24) where q N 2 2 1/ 2 (k l ) . 2f (3.5.25) The solution of (3.5.24) of can be written P( z ) A sinh( 2qz ) B cosh( 2qz ) , (3.5.26) where A and B are constants of integration. By applying (3.5.22) at z = 0, we obtain that B ( 2qcf / g ) A , i.e. 2qcf P( z ) Asinh( 2qz ) cosh( 2qz ) g (3.5.27) By inserting (3.5.27) into (3.5.22) for z = H, we obtain an equation for the complex phase speed c: 4q 2 f 2 g 2qH coth( 2qH ) 1 0 . c 4q 2 Hc g f (3.5.28) After some algebra, we find that the solution to this equation can be written c U max 2 1 1/ 2 1 q* coth q* 1q* tanh q* 1 , q* (3.5.29) where q* = qH. Let c cr ici in (3.5.23). Accordingly, if ci > 0, the perturbation solution will grow exponentially in time, which means that the basic state is unstable. Oppositely, the basic state is stable if ci < 0. Since always coth q* 1/ q* for q* 0 , it is obvious from (3.5.29) that a root where ci > 0, requires that q* tanh q * 1 i.e. q* q*c 1.2 . (3.5.30) Inserting from (3.5.25), the condition for instability can be written N 2H 2 2 2 (k l ) 4q*2c 5.76 . f2 Unstable waves that satisfy (3.5.31), are usually referred to as Eady waves. (3.5.31) 105 It is generally accepted that observed eddies in current systems like the Gulf Stream and the Norwegian Coastal Current (NCC) may be caused by baroclinic instability. Let us use NCC as an example. In a lecture note, Aas (1985) summarizes some of the known facts about NCC. With reference to the parameters in the present stability analysis, we can assume that H = 100 m and L = 40 km. Furthermore, we take that the surface value of t is 23 close to the coast, and 27 where the coastal water borders the Atlantic inflow. This yields = 107m1 in equation (3.5.1). Insertion into (3.5.2), with f = 1.3104s1, yields a surface current Umax= 0.76 m s1, i.e. a mean northward flow of 0.38 m s1. Furthermore, if t varies from 23 to 27 over a vertical distance of 100 m, then N = 2102s1 in (3.5.18). We consider the first mode (n = 1) in the y-direction, which means that l = /L. We calculate the non-dimensional growth rate kci / f from (3.5.29) as a function of the wavelength the in the x-direction by using the values assessed here. The result is depicted in Fig. 3.9. Fig. 3.9 Non-dimensional growth rate as a function of the wavelength for Eady waves. We observe from the figure that maximum growth occurs for waves with 80 km. This scale fits well with observed length scales for meanders in NCC (Aas, 1985). For the most unstable Eady wave with = 80 km, (3.5.29) yields for the complex phase speed in this example: c U max (0.5 0.17i ) . (3.5.32) 106 The time Te it takes before the amplitude of the disturbance increases by a factor e (the e-folding time) is given by Te 1 . kci (3.5.33) The numerical value of the e-folding time for the most unstable wave in this example is Te 27 hrs. The period for this wave, T 2 /( kcr ) , is here approximately 58 hrs. We note that these time scales are sufficiently large to justify the application of the quasi-geostrophic approximation, which requires that the time scale of the motion should be considerably larger than the inertial period 2 /f (13 hrs in this example). The growth rate of the disturbances calculated here is comparable with the observed time scales for the development of meanders in NCC (Aas, 1985). REFERENCES Articles Charney, J. G., and Eliassen, A.: 1949 Tellus, 1, 38. Eady, E. A.: 1949 Tellus, 1, 33. Eide, L. I.: 1979 Deep-Sea Res., 26, 601. Gill, A. E., and Clarke, A. J.: 1974 Deep-Sea Res., 21, 325. 1979 J. Phys. Oceanogr., 9, 1126. 1981 Tellus, 33, 402. Munk, W.: 1950 J. Met., 7, 79. Pedlosky, J.: 1965 J. Mar. Res., 23, 207. Stommel, H.: 1948 Trans. Am. Geophys. Un., 29, 202. Sverdrup, H. U.: 1927 Geophys. Publ., 4, 75. Sverdrup, H. U.: 1947 Proc. Nat. Acad. Sci., Wash., 33, 318. Taylor, G. I.: 1916 Proc. Roy. Soc., Ser. A, XC 11, 196. Taylor, G.I.: 1922 Proc. Roy. Soc., Ser. A, 102, 180. Aas, E.: 1985 Lecture note, 27 pp. Martinsen, E. A, Gjevik, B., and Røed, L. P.: Martinsen, E. A., and Weber, J. E.: 107 Books Defant, A.: 1961 Physical Oceanography, Vol. I & II. Pergamon Press 1961. Gill, A. E.: 1982 Atmosphere-Ocean Dynamics. Academic Press 1982. Krauss, W.: 1973 Methods and Results of Theoretical Oceanography. Vol. I Gebrüder Borntraeger 1973. LeBlond, P. H., and Mysak, L. A.: 1978 Waves in the Ocean. Elsevier 1978. Pedlosky, J.: 1987 Geophysical Fluid Dynamics, 2. ed. Springer 1987.