Kei Sakamoto and Kazunori Akitomo Department of Geophysics ... · Kei Sakamoto and Kazunori...

1
0 1 2 3 4 5 z -1.0 0.0 1.0 utide 0 1 2 3 4 5 z -1.0 0.0 1.0 utide -1.0 0.0 1.0 utide 0 2 4 6 8 10 time (tidal cycle) 10 -35 10 -30 10 -25 10 -20 10 -15 10 -10 10 -5 10 0 EKE v 0.0 0.5 1.0 1.5 2.0 Ro t 0 2 4 6 8 10 Tidal amplutude (cm/s) Lx free slip no slip x z y tidal current U tide cos t O f / 2 H Ly 0 20 40 60 80 100 120 x 0 5 10 15 z 0 20 40 60 80 100 120 x 0 20 40 60 80 100 120 y 0 20 40 60 80 100 120 y 0 5 10 15 z 0 20 40 60 80 100 120 x 0 20 40 60 80 100 120 y 0 1 2 3 4 5 z 0 1 1.1e-21 1.5e-21 6.8e-22 4.5e-19 1.1e-17 mean 0 1 2 3 4 5 z 0 1 0 1 2 3 4 5 z 0 1 2.4e-18 3.9e-23 1.6e-16 7.7e-09 2.3e-08 mean 0 1 2 3 4 5 z 0 1 0 1 2 3 4 5 z 0 1 1.7e-20 5.2e-20 1.3e-19 1.8e-19 1.4e-19 mean 0 1 2 3 4 5 z 0 1 x y z u,w v -1.0 -0.5 0.0 0.5 1.0 0 2 4 6 8 10 z -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 4 5 6 z 0 Instabilities of the tidally-induced bottom boundary layer in the rotating frame Kei Sakamoto and Kazunori Akitomo, Department of Geophysics, Kyoto University e-mail: [email protected] 1. Introduction Shear instability of the tidally-induced bottom boundary layer (TBBL) contributes to water mixing in the ocean. Recent observations and model studies showed that this contribution is especially important in polar oceans, where the inertial and tidal periods are close (Nost 1994, Furevik and Foldvik 1996, Makinson 2002). However, little is known with regard to the dynamics of the shear instability in the TBBL under influence of the Earth's rotation. On the other hand, many studies have focused on instabilities of the Ekman layer (the boundary layer induced by steady flow in the rotating frame) and the Stokes layer (by oscillating flow without rotation). The Ekman layer is susceptible to two types of instability. The type I instability: Perturbation energy is directly supplied to u from inflectional points of the x-component of the basic flow. The type II instability: Energy is supplied to v from shear of the y- component of the basic flow and redistributed to u by the Coriolis effect. In either case, two-dimensional perturbation rolls appear within the boundary layer . The Stokes layer becomes unstable when the Reynolds number exceeds approximately 550. Since inflectional points in the vertical profile of the basic flow move upward, perturbation rolls develop mainly above the boundary layer in contrast to the Ekman layer. The TBBL in the actual ocean is expected to be susceptible to these three instabilities. The purpose of this study is to investigate what dynamics control the stability of the TBBL in the rotating frame. We executed numerical experiments with some values of the temporal Rossby number, Rot = the inertial period / the tidal period, which indicates the ratio of the effects of oscillation and rotation. 2. Numerical model Model basin: Governing equations: The non-hydrostatic equation of motion and the equation of continuity under the rigid-lid approximation. The variables are derived into the basic flow (vtide) and perturbation (velocity v, pressure p). Boundary conditions: Surface: Rigid-lid and free-slip ( u/ z= v/ z=w=0) Bottom: no-slip (u=v=w=0) Horizontal boundaries: cyclic Initial condition: Minute random perturbation Non-dimensionalization: The governing equations are non-dimensionalized by the vertical scale of the TBBL, Htide, and the amplitude of tidal flow, Utide. Size of the model basin and grid intervals: Lx=Ly=128, H=64, x= y=1.0, z=0.1 10 (100 grids) Experimental cases and basic flow: To focus on dependence on the temporal Rossby number, we fix the tidal amplitude (8.53cm/s). Differences 1: Htide is large when Rot 1. 2: The flow direction is reversed downward when Rot < 1 while upward when Rot > 1. 3. Results All the cases are unstable. We analyze periods during which EKE increases exponentially, indicating the "linear stage". 3.1 Perturbation In all the cases, two-dimensional rolls develop. In cases A and B, the rolls are within the boundary layer (z<3), which is same as the Ekman layer. In cases C and D, the rolls are above that (z>3) like the Stokes layer. 3.2 The EKE equation To distinguish the type II instability from the type I and Stokes instabilities (i.e. inflectional instabilities), we analyze the EKE equation in the (x, z) plane after Kaylor and Faller (1972), where x (y) is the direction along with the rolls have structure (no structure). 3.3 Supplemental experiments We executed numerical experiments covering wide ranges of Rot and the tidal amplitude, using a vertical two-dimensional model, and investigated stability of the TBBL. 1. The type of instability mainly depends on Rot. (Details are desribed below) 2. The TBBL is unstable even at the small tidal amplitude when 0.9 < Rot < 1.1, which is qualitatively consistent with previous ocean observations. This is due to increase of the vertical scale Htide (Difference 1 of the basic flow), leading to the large Reynolds number. 3.4 Discussion on the dependence of the instability dynamics The type II instability when 1.0 < Rot < 1.1 Scaling analysis of the EKE equation shows that the Coriolis term / the non-linear term ~ f Htide / Utide. Therefore, the type II instability is preferable in this range of Rot, where Htide is remarkably increased. However, this effect works only when Rot > 1.0, where the flow direction is reversed upward and the thick shear layer is maintained (Difference 2). The type I instability when Rot < 1.0 The type II instability is depressed since the shear layer becomes thin at once due to the downward movement of the tidal reversal (Difference 2). The Stokes instability when Rot > 1.1 The effect of oscillation is predominant. The tidal amplitude at which the TBBL becomes unstable is identical with that of the Stokes layer. 4. Summary To investigate instability of the bottom boundary layer induced by tidal flow (oscillating flow) in a rotating frame, numerical experiments have been carried out with a three-dimensional non-hydrostatic model. Three types of instability are found depending on the temporal Rossby number Rot, the ratio of the inertial and tidal periods. When Rot < 1.0, the Ekman type I instability occurs because the effect of rotation is dominant. When Rot > 1.1, the Stokes layer instability is excited as in the absence of rotation. When 1.0 < Rot < 1.1, the thickening of the boundary layer enhances the contribution of the Coriolis effect to destabilization, so that the Ekman type II instability peculiar to a rotating frame tends to appear. We are now studying turbulence of the TBBL to clarify mixing effect of these instabilities. Figure 1 A schematic picture on instability of the Ekman layer. (u, v, w) is perturbative motion induced by instability. The y-axis is set to the direction along which perturbation has no structure. Basic flow in the Ekman layer. Figure 2 Basic flow in the Stokes layer. : inflectional point A three-dimensional basin of the size of Lx Ly H. Figure 3 Model basin and the coordinate system. Figure 4 Vertical profiles of basic flow. Figure 5 Time evolution of the areal average of the eddy kinetic energy, EKE. Figure 6 Instantaneous distributions of the vertical velocity (w) Figure 7 Time evolutions of the terms of the EKE equation and basic flow. Case A The non-linear term is predominant. Taking account of the characteristics of perturbations, we can regard this instability as the type I instability . The positive non-linear term coincides with the inflectional point of utide. We have similar results in case B (not shown). Figure 8 Types of instability. :stable :type I instability :type II instability :Stokes instability :mixture of type I and II instabilities The results of the 3D experiments are also shown. The non-linear term: the type I or Stokes instabilities The Coriolis term: the type II instability The dissipation term The diffusion term (u, v) and (utide, vtide): (x, y) components of v and vtide utide vtide normalized by time utide vtide utide vtide indicate the inflectional points (i.e. local maxima of shear) Case C The Coriolis term is predominant. The type II instability EKE is supplied in thick shear layer of vtide. Case D The non-linear term is predominant. The Stokes instability , taking account of the perturbation shown in Fig. 6. x-component y-component time f: Coriolis parameter 0: standard density (1.027g/cm 3 ) : eddy viscosity (1cm 2 /s) Case: Rot: Htide: Reynolds number (Htide Utide / ) Ekman 0 1.2m 1000 A 0.5 1.2m 1000 B 0.95 5.1m 4350 C 1.05 5.4m 4580 D 2.0 1.7m 1410 Stokes 1.2m 1000 time a indicates horizontal average of a. x,y Cases: Ekman A, B, C, D Stokes (dashed) up down Case A Case D Horizontal Vertical Case A B C D Height of the boundary layer case A C D

Transcript of Kei Sakamoto and Kazunori Akitomo Department of Geophysics ... · Kei Sakamoto and Kazunori...

Page 1: Kei Sakamoto and Kazunori Akitomo Department of Geophysics ... · Kei Sakamoto and Kazunori Akitomo, Department of Geophysics, Kyoto University ... Perturbation energy is directly

0

1

2

3

4

5

z

-1.0 0.0 1.0utide

0

1

2

3

4

5

z

-1.0 0.0 1.0utide

-1.0 0.0 1.0utide

0 2 4 6 8 10time (tidal cycle)

10-35

10-30

10-25

10-20

10-15

10-10

10-5

100

EK

Ev

0.0 0.5 1.0 1.5 2.0R ot

0

2

4

6

8

10

Tid

al a

mp

lutu

de

(cm

/s)

Lx

free slip

no slipx

z y

tidal current U

tide cos t

O

f / 2

H

Ly

0 20 40 60 80 100 120x

0

5

10

15

z

0 20 40 60 80 100 120x

0

20

40

60

80

100

120

y

0 20 40 60 80 100 120y

0

5

10

15

z

0 20 40 60 80 100 120x

0

20

40

60

80

100

120

y

0

1

2

3

4

5

z

0 1

1.1e-21

1.5e-21

6.8e-22

4.5e-19

1.1e-17

mean

0

1

2

3

4

5

z

0 1

0

1

2

3

4

5

z

0 1

2.4e-18

3.9e-23

1.6e-16

7.7e-09

2.3e-08

mean

0

1

2

3

4

5

z

0 1

0

1

2

3

4

5

z

0 1

1.7e-20

5.2e-20

1.3e-19

1.8e-19

1.4e-19

mean

0

1

2

3

4

5

z

0 1

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

x

y

z

u,w

v

-1.0 -0.5 0.0 0.5 1.00

2

4

6

8

10

z

-1.0 -0.5 0.0 0.5 1.00

1

2

3

4

5

6

z

0

Instabilities of the tidally-induced bottom boundary layer in the rotating frameKei Sakamoto and Kazunori Akitomo, Department of Geophysics, Kyoto University

e-mail: [email protected]

1. Introduction

Shear instability of the tidally-induced bottom boundary layer (TBBL) contributes to water mixing in the ocean. Recent observations and model studies showed that this contribution is especially important in polar oceans, where the inertial and tidal periods are close (Nost 1994,

Furevik and Foldvik 1996, Makinson 2002). However, little is known with regard to the dynamics of the shear instability in the TBBL under influence of the Earth's rotation.

On the other hand, many studies have focused on instabilities of the Ekman layer (the boundary layer induced by steady flow in the rotating frame) and the Stokes layer (by oscillating flow without rotation). The Ekman layer is susceptible to two types of instability.

The type I instability: Perturbation energy is directly supplied to u from inflectional points of the x-component of the basic flow.The type II instability: Energy is supplied to v from shear of the y-component of the basic flow and redistributed to u by the Coriolis effect. In either case, two-dimensional perturbation rolls appear within the boundary layer.

The Stokes layer becomes unstable when the Reynolds number exceeds approximately 550. Since inflectional points in the vertical profile of the basic flow move upward, perturbation rolls develop mainly above the boundary layer in contrast to the Ekman layer.

The TBBL in the actual ocean is expected to be susceptible to these three instabilities. The purpose of this study is to investigate what dynamics control the stability of the TBBL in the rotating frame.We executed numerical experiments with some values of the temporal Rossby number, Rot = the inertial period / the tidal period,which indicates the ratio of the effects of oscillation and rotation.

2. Numerical model

Model basin:

Governing equations: The non-hydrostatic equation of motion and the equation of continuity under the rigid-lid approximation. The variables are derived into the basic flow (vtide) and perturbation (velocity v, pressure p).

Boundary conditions: Surface: Rigid-lid and free-slip (∂u/∂z=∂v/∂z=w=0) Bottom: no-slip (u=v=w=0) Horizontal boundaries: cyclic

Initial condition: Minute random perturbation

Non-dimensionalization: The governing equations are non-dimensionalized by the vertical scale of the TBBL, Htide, and the amplitude of tidal flow, Utide.

Size of the model basin and grid intervals: Lx=Ly=128, H=64, ⊿x=⊿y=1.0, ⊿z=0.1~10 (100 grids)

Experimental cases and basic flow: To focus on dependence on the temporal Rossby number, we fix the tidal amplitude (8.53cm/s).

Differences 1: Htide is large when Rot~1. 2: The flow direction is reversed downward when Rot < 1 while upward when Rot > 1.

3. Results

All the cases are unstable. We analyze periods during which EKE increases exponentially, indicating the "linear stage".

3.1 Perturbation

In all the cases, two-dimensional rolls develop.In cases A and B, the rolls are within the boundary layer (z<3), which is same as the Ekman layer.In cases C and D, the rolls are above that (z>3) like the Stokes layer.

3.2 The EKE equation

To distinguish the type II instability from the type I and Stokes instabilities (i.e. inflectional instabilities), we analyze the EKE equation in the (x, z) plane after Kaylor and Faller (1972), where x (y) is the direction along with the rolls have structure (no structure).

3.3 Supplemental experiments

We executed numerical experiments covering wide ranges of Rot and the tidal amplitude, using a vertical two-dimensional model, and investigated stability of the TBBL.

1. The type of instability mainly depends on Rot. (Details are desribed below)

2. The TBBL is unstable even at the small tidal amplitude when 0.9 < Rot < 1.1, which is qualitatively consistent with previous ocean observations.This is due to increase of the vertical scale Htide (Difference 1 of the basic flow), leading to the large Reynolds number.

3.4 Discussion on the dependence of the instability dynamics

The type II instability when 1.0 < Rot < 1.1 Scaling analysis of the EKE equation shows that the Coriolis term / the non-linear term ~ f ×Htide / Utide.Therefore, the type II instability is preferable in this range of Rot, where Htide is remarkably increased. However, this effect works only when Rot > 1.0, where the flow direction is reversed upward and the thick shear layer is maintained (Difference 2).

The type I instability when Rot < 1.0 The type II instability is depressed since the shear layer becomes thin at once due to the downward movement of the tidal reversal (Difference 2).

The Stokes instability when Rot > 1.1 The effect of oscillation is predominant. The tidal amplitude at which the TBBL becomes unstable is identical with that of the Stokes layer.

4. Summary

To investigate instability of the bottom boundary layer induced by tidal flow (oscillating flow) in a rotating frame, numerical experiments have been carried out with a three-dimensional non-hydrostatic model. Three types of instability are found depending on the temporal Rossby number Rot, the ratio of the inertial and tidal periods.

When Rot < 1.0, the Ekman type I instability occurs because the effect of rotation is dominant.When Rot > 1.1, the Stokes layer instability is excited as in the absence of rotation.When 1.0 < Rot < 1.1, the thickening of the boundary layer enhances the contribution of the Coriolis effect to destabilization, so that the Ekman type II instability peculiar to a rotating frame tends to appear.

We are now studying turbulence of the TBBL to clarify mixing effect of these instabilities.

Figure 1 A schematic picture on instability of the Ekman layer.(u, v, w) is perturbative motion induced by instability.

The y-axis is set to the direction along which perturbation has no structure.

Basic flow in the Ekman layer.

Figure 2 Basic flow in the Stokes layer. : inflectional point

A three-dimensional basin of the size of Lx×Ly×H.

Figure 3 Model basin and the coordinate system.

Figure 4 Vertical profiles of basic flow.

Figure 5 Time evolution of the areal average of the eddy kinetic energy, EKE.

Figure 6 Instantaneous distributions of the vertical velocity (w)

Figure 7 Time evolutions of the terms of the EKE equation and basic flow. Case A

The non-linear term is predominant. Taking account of the characteristics of perturbations, we can regard this instability as the type I instability. The positive non-linear term coincides with the inflectional point of utide. We have similar results in case B (not shown).

Figure 8 Types of instability.

×:stable □:type I instability△:type II instability *:Stokes instability◇:mixture of type I and II instabilities

The results of the 3D experiments are also shown.

The non-linear term: the type I or Stokes instabilities

The Coriolis term: the type II instability

The dissipation term

The diffusion term

(u, v) and (utide, vtide): (x, y) components of v and vtide

utide

vtide

normalized by

time

utide

vtide

utide

vtide

◇ indicate the inflectional points (i.e. local maxima of shear)

Case CThe Coriolis term is predominant. ⇒ The type II instabilityEKE is supplied in thick shear layer of vtide.

Case DThe non-linear term is predominant. ⇒ The Stokes instability, taking account of the perturbation shown in Fig. 6.

x-component

y-component

time

f: Coriolis parameterρ0: standard density (1.027g/cm3)ν: eddy viscosity (1cm2/s)

Case:Rot:Htide:Reynolds number(Htide Utide / ν)

Ekman01.2m1000

A0.51.2m1000

B0.955.1m4350

C1.055.4m4580

D2.01.7m1410

Stokes ∞1.2m1000

time

a indicates horizontal average of a.x,y

Cases: EkmanA, B, C, D

Stokes (dashed)

updown

Case A Case DHorizontal

Vertical

Case A B C D

Height of the boundary layer

case A

C

D