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EVAT 554OCEAN-ATMOSPHERE
DYNAMICS
TIME-DEPENDENT DYNAMICS; WAVE DISTURBANCES
LECTURE 21
“Buoyancy Waves”
Recall the vertical momentum balance for a nearly incompressible fluid that is perturbed from its initial state
02
12/2
zscg
zgdtzd
022/2 zNdtzd
)cos(NtAz
More generally we have travelling wave disturbances...
Represents vertical oscillations due to restoring force of gravity, given some initial perturbation
(eventually damped by friction)
“Gravity Waves”
Lateral pressure gradients arise from the perturbed free surface:
xgxp // ygyp //
xpftu
/v
Consider a perturbation from geostrophic balance
ypft
/uv
)/v/(/ yxuht The continuity equation takes the approximate form (for small h):
More generally we have travelling wave disturbances...
“Gravity Waves”
Lateral pressure gradients arise from the perturbed free surface:
xgxp // ygyp //
xpftu
/v
ypft
/uv
)/v/(/ yxuht The continuity equation takes the approximate form (for small h):
xgftu
/v
ygft
/uv
Consider a perturbation from geostrophic balance
“Gravity Waves”
)/v/(/ yxuht The continuity equation takes the approximate form (for small h):
xgftu
/v
ygft
/uv
Assume the scale of motion is small compared to the planetary scale
)/v/(/22
tytxuht
22/ xg
txu
22/v yg
ty
xgtu
/ yg
t
/v
Differentiate these expressions,
)//(/222222yxght
Consider a perturbation from geostrophic balance
“Gravity Waves”
)//(/222222yxght
This is the equation of a traveling wave!
For simplicity, assume that the free surface gradient is non-zero only along the x direction
)/(/2222xght
0)/(/22222 xct
The solution to this equation is:
)cos(0
tkx
ghc
Shallow Water waves
ghkkcDispersion relation
/2/2 Lnk,...2,1n
“Gravity Waves”
This is the equation of a traveling wave!
For simplicity, assume that the free surface gradient is non-zero only along the x direction
)/(/2222xght
0)/(/22222 xct
The solution to this equation is:
)cos(0
tkx
ghc
ghkkc
)cos(0 tkxgk
u
Dispersion relation
/2/2 Lnk,...2,1n
Shallow Water waves
xgtu
/ )sin(
0tkxgk
“Gravity Waves”
xgtu
/ )sin(
0tkxgk )cos(0 tkx
gku
Shallow Water waves
Planetary (“Rossby”) Waves”
xpftu
/v
Consider once again a perturbation from geostrophic balance
ypft
/uv
τz
pf2
u
V
Relative Vorticityu21
2
fzffp
fzff
pfa
τ
12 Absolute Vorticity
τzff
p12
Recall from an earlier lecture
Planetary (“Rossby”) Waves”
τzff
p
12
fp
k
ˆ)(
2V
This gives an expression for the Vorticity in the absence of any frictional stresses
xpftu
/v
Consider once again a perturbation from geostrophic balance
ypft
/uv
Planetary (“Rossby”) Waves”
τzff
p
12
fp
k
ˆ)(
2V
This gives an expression for the Vorticity in the absence of any frictional stresses
xpftu
/v
Consider once again a perturbation from geostrophic balance
ypft
/uv
Define the streamfunctionfp
ˆ
x
v'y
u'
Planetary (“Rossby”) Waves”
fp
k
ˆ)(
2V
This gives an expression for the Vorticity in the absence of any frictional stresses
0y)(dtd
0y)2(dtd
Conservation of absolute vorticity on a beta plane, gives
Define the streamfunctionfp
ˆ
x
v'y
u'
Planetary (“Rossby”) Waves”
Define the streamfunctionfp
ˆ
Conservation of absolute vorticity on a beta plane, gives
x v'
y u'
linearize under the assumption of a constant zonal flow
xu
tdt
d
0y)(dtd
0y)2(dtd
0)2()2(t
xxu
The solution has the form of a traveling wave:
)]}(exp[Re{0
tlykxi
)]}(exp[Re{u'0
tlykxiily
)]}(exp[Re{v'0
tlykxiikx
Planetary (“Rossby”) Waves”
Define the streamfunctionfp
ˆ
x
v'y
u'
0)2()2(t
xxu
The solution has the form of a traveling wave:
Plugging the traveling wave solution into the equation gives,
22 lkuc
kc Dispersion Relation
)]}(exp[Re{0
tlykxi
If the meridional velocity field represents a geostrophically-balanced standing wave perturbation of the free surface xgf /v
0
Then we haveghfk
uc/2
0
2
)]}(exp[Re{u'0
tlykxiily
)]}(exp[Re{v'0
tlykxiikx
Planetary (“Rossby”) Waves”
Plugging the traveling wave solution into the equation gives,
22 lkuc
kc Dispersion Relation
)]}(exp[Re{u'0
tlykxiily
)]}(exp[Re{v'0
tlykxiikx
If the meridional velocity field represents a geostrophically-balanced standing wave perturbation of the free surface xgf /v
0
Then we haveghfk
uc/2
0
2
Planetary (“Rossby”) Waves”
Rossby Radius||/0fghr
If h=1 km, r1500km
ghfkuc
/20
2
kc Dispersion Relation
Planetary (“Rossby”) Waves”
The periods of Rossby Waves in the Ocean that are
possible is Determined by Latitude and Basin Width
“Kelvin Waves”
Lateral pressure gradients arise from the perturbed free surface:
xgxp // ygyp //
xpftu
/v
Consider again a perturbation from geostrophic balance
ypft
/uv
The continuity equation takes the approximate form (for small h):
)/v/(/ yxuht
“Kelvin Waves”
)/v/(/ yxuht The continuity equation takes the approximate form (for small h):
xgftu
/v
ygft
/uv
Do not assume that the scale of motion is small compared to the planetary scale
xgftu
/v ygf
t
/uv
Consider again a perturbation from geostrophic balance
Consider an east-west boundary
ygf /u v=0 v/ t =0
xgtu // 22// xgtxu
xuh /
txuht // 22 22/ xgh
)/(/2222xght
“Kelvin Waves”
)/(/2222xght
)cos()(0
tkxy The solution is ghkkc
xuht // xuhtkxy /)sin()(0
htkxyxu /)sin()(/
0
khtkxyu /)cos()(0
ygf /u )cos(/)(0
/)cos()(0
tkxyygkhtkxyf
)/v/(/ yxuht xuh /
“Kelvin Waves”
)/(/2222xght
)cos()(0
tkxy The solution is ghkkc
xuht // xuhtkxy /)sin()(0
htkxyxu /)sin()(/
0
khtkxyu /)cos()(0
ygf /u )cos(/)(0
/)cos()(0
tkxyygkhtkxyf
Thus we have
yygkhyf /)(0
/)(0
ghfyy /exp)(00
)cos(/exp0
tkxghfy
“Kelvin Waves”
)cos(/exp0
tkxghfy
khtkxghfyu /)cos(/exp0
This also generalizes to equatorially-trapped waves!
These are “coastally-trapped” waves
Development would be identical for North-South boundary where u=0
Length scale is Rossby Radiusf
ghr
Equatorial Radius of Deformation:
2/1ghr
“Kelvin Waves”
)cos(/exp0
tkxghfy
khtkxghfyu /)cos(/exp0
The free surface () can be interpreted in terms of the mean depth of the thermocline
“Kelvin Waves”
)cos(/exp0
tkxghfy
khtkxghfyu /)cos(/exp0
The free surface () can be interpreted in terms of the mean depth of the thermocline
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