The Basic State

In an extension to previous work we investigate the effects of finite deformation length on barotropic instability in a critical layer. Initially we considered the same

−4πµ2 _{0 4πµ}2 U(y) −π 0 π y/2 µ L D = 10 L D = 2 L D = 1 L D = 0.5 L_D = 0.25 L D = 0.125 L D = 0.0625

Figure 2.7: U(y) for a given shear and 1

16 ≤LD ≤10.

background linear shear flow, U(y) = Λy, from the infinite deformation length experiments. However the potential vorticity associated with the shear flow for finiteLD is

q=βy+ Λ 1₂LD−2y2−1

(2.9) which is a quadratic in y. As LD −→ 0 the quadratic in (2.9) is dominated by

the LD term and becomes non-monotonic, i.e. unstable. Therefore a uniform

shear flow has non-monotonic potential vorticity and this is undesirable because the evolution of the critical layer will be obscured by the possible instability of the background flow itself.

As a possible alternative, we consider the shear arising from a uniform poten- tial vorticity basic state q =∇2_Ψ−L

D−2Ψ = 0. This along with the condition

of no net momentum in any direction:

Z 2πΛ

−2πΛ

¯

udy= 0 (2.10)

U(y) = A LD sinh y LD (2.11) whereAis a constant. Subsequently we examined two different conditions to find the constant A. The first condition was to define the shear at the center of the channel (y = 0) to be the same as the shear profiles investigated in the infinite deformation length experiments. This resulted in A = ΛLD2 where uy |y=0= Λ.

The other condition considered was that the velocity difference across the channel was the same as in the LD−1 = 0 cases. This gave

A = Λ2πΛLD sinh(2_LπΛ

D )

(2.12) where y = ±2πΛ represents the channel walls. However neither of these condi- tions were suitable for our investigation. In the case of the first condition the velocities at the channel walls were too large at small LD for the numerical cal-

culation due to the hyperbolic sine term in (2.11). For the second condition the width of the critical layer equalled the width of the channel at small LD due to

the small velocities across the channel. This then caused uncertainties over the effects of the boundaries on the evolution.

Ultimately we decided that the average shear over the width of the critical layer should be comparable to the shear over the critical layer in the infinite Rossby deformation length experiments. We defined a top hat weighting

g(y) =    1 2yc if |y|< yc 0 otherwise. (2.13)

whereyc= 2₅(2πΛ) denotes the edge of the critical layer (critical layer is 2₅ of the

channel width), such that the average shear over the critical layer is equal to Λ as follows

(a)

(b)

(c)

(d)

(e)

Figure 2.8: Snapshots of potential vorticity for the case Λ = 1

8 att= 1.2,1.4,1.6

for (a) LD = 10, (b) LD = 2 and (c) LD = 1, at t = 1,1.2,1.4 for (d) LD = 1₂ and att= 0.8,1,1.2 for (e) LD = 1

4. Note: only the middle half of the domain is

Z l −l g(y)uydy = Λ (2.14) Z yc −yc uy 1 2yc dy = Λ (2.15) 1 2yc (u(yc)−u(−yc)) = Λ (2.16)

The constant A in (2.11) is then defined from (2.14) to be

A= 2ycLDΛ [sinh(yc LD)−sinh( −yc LD)] (2.17) The background flow, U(y) (2.11), with A as defined above is shown in Fig- ure 2.7 for a given shear and 1

16 ≤ LD ≤ 10. The average shear in the critical

layer (i.e. between y=±2₅(2πµ)) is equal to the shear Λ (=µ).

In the infinite Rossby deformation length experiments we defined a topo- graphic streamfunction (ψtopo, ( 2.8)) that coincided with the perturbation stream-

function of the SWW solution (1978). In these experiments whereLDis finite we

specify a topographic forcing qtopo such that ψtopo is the same as the infinite LD

cases. This yields

qtopo = ∇2ψtopo−LD−2ψtopo (2.18)

= −ǫµcos(kx) 1 +LD−2 1− coshy cosh(2πµ) (2.19)

It is important here that the streamline pattern of the background flow and the topography is the same closed streamline pattern of the infiniteLD experiments.

The reason for this is that the streamline pattern shows the relative importance of ¯u(0) and the topographic forcing and we want this to be constant across all Λ and LD.

Figure 2.8 shows snapshots of the critical layer potential vorticity for Λ = 1₈ at different times for different values ofLD. Figure 2.8 demonstrates that when LD

is small (LD <1) the width of the critical layer increases. This is an interesting

result because we have set the problem up so that as far as possible everything is the same as the infinite LD cases (topographic forcing, average shear in the

critical layer region) and despite this we have found that the critical layer width increases at smallLD. The increase in critical layer width is due to the decrease in LD and its consequent affect on the Rossby wave elasticity. WhenLDis relatively

large (Figure 2.8 (a), (b) and (c)) and we perturb the potential vorticity contours, the contours resist the motion due to Rossby wave elasticity and are therefore difficult to deform. In contrast at smallerLD (Figure 2.8 (d) and (e)) the Rossby

wave elasticity is weakened and hence there is less resistance to the deformation of potential vorticity contours resulting in a wider critical layer. Note that the increase in critical layer width is not due to a resonance with the topography (as occurred in the infinite deformation length cases). Other experiments at smaller

LD showed further increases in the critical layer width indicating that this is a

trend (not resonance). Also the time evolution of the critical layers in these cases does not show linear growth in the critical layer width.