2009 C5: Physics of Atmospheres and Oceans
5. The equations of motion for a two-dimensional ocean of uniform depth, forced by a zonal wind stress and
dissipated by linear friction, can be written:
∂u ∂u ∂u 1 ∂p τs
+u +v − fv + = − ru,
∂t ∂x ∂y ρ0 ∂x ρ0 H
∂v ∂v ∂v 1 ∂p
+u +v + fu + = −rv,
∂t ∂x ∂y ρ0 ∂y
∂u ∂v
+ = 0,
∂x ∂y
where the symbols have their usual meanings.
Derive the vorticity equation:
∂ ∂ ∂ 1 ∂τs
+u +v ζ =− − rξ,
∂t ∂x ∂y ρ0 H ∂y
where ζ is the absolute vorticity and ξ is the relative vorticity.
Taking the curl of the momentum equations gives:
∂ ∂ ∂ ∂u ∂v ∂u ∂v df 1 ∂τs
+u +v ξ+ξ + +f + +v =− − rξ,
∂t ∂x ∂y ∂x ∂y ∂x ∂y dy ρ0 H ∂y
where the relative vorticity is defined as
∂v ∂u
ξ= − ,
∂x ∂y
and we assume f = f (y) (although the result holds for arbitrary f ).
Using the continuity equation to eliminate two terms in the above, and defining the absolute vorticity
as ζ = ξ + f , the result follows:
∂ ∂ ∂ 1 ∂τs
+u +v ζ =− − rξ.
∂t ∂x ∂y ρ0 H ∂y
[5]
Write down the form of the vorticity equation known as “Sverdrup balance”, which holds approximately in
the interior of an ocean gyre. List the conditions required for its validity. Assuming that the wind stress
varies only with latitude, show that the northward volume transport predicted by Sverdrup balance is
L ∂τs
T =− ,
βρ0 ∂y
where L is the basin width and β has its usual meaning. Sketch the circulation predicted by Sverdrup balance
in a typical hemispheric ocean basin, stating clearly any boundary conditions that you assume. Estimate the
volume transport of a typical subtropical gyre, assuming a maximum wind stress of magnitude 0.2 N m−2 and
typical values for the remaining parameters.
The approximate Sverdrup balance is between advection of planetary vorticity and the source of vorticity
from the wind stress curl:
1 ∂τs
βv = − ,
ρ0 H ∂y
where β = df /dy.
This requires neglect of: relative vorticity, assumed small compared with variations in planetary vorticity; the
frictional vorticity sink, small outside the boundary currents; and time-dependence, negligible on time-scale
longer than the barotropic Rossby wave propagation time-scale. (There is no need to give mathematical
conditions for validity of these approximations.)
The northward transport predicted by Sverdrup balance is
L ∂τs
Z
Hv dx = − ,
βρ0 ∂y
as required.
9
,2009 C5: Physics of Atmospheres and Oceans
Assume easterly trades in the tropics, westerlies at mid-latitudes and easterlies (or at least weakening
westerlies) at high latitudes.
Also assume no flow through the eastern boundary, following Sverdrup’s original calculation, on the
basis that we know boundary currents form at the western margins of gyres where Sverdrup balance breaks
down.
Thus (please ignore Ψ labels):
polar
easterlies
subpolar
Ψ− ocean
mid−latitude
westerlies
subtropical
Ψ+ ocean
trades
τs
Gyre transport corresponds to maximum equatorward Sverdrup transport, as evident from the previous
figure.
Estimate wind stress curl as change in wind stress between equator and mid-latitudes (say 40◦ N):
∂τ 0.4 N m−2
∼ = 10−7 N m3 .
∂y 4 × 106
Take the width of the Atlantic (say) to be 5 000 km and β ≈ 2 × 10−11 m−1 s−1 and ρ0 ∼ 103 kg m−3 .
Thus
L ∂τs −1
Tgyre ≈ ∼ 2.5 × 107 m3 s (= 25 Sv).
βρ0 ∂y max
(Give full credit for any answer in the right ballpark here.)
[10]
10
, 2009 C5: Physics of Atmospheres and Oceans
Suppose that the gyre is closed by a frictional boundary current of width δ. By scaling the appropriate terms
in the vorticity equation, or otherwise, deduce that
r
δ∼ .
β
Assuming a frictional time scale of 1 year and H = 500 m, estimate δ and a typical boundary current velocity.
Comment on the extent to which these values are realistic. Using these values, estimate the magnitude of
the relative vorticity in the boundary current, and thus deduce that it cannot be neglected in reality.
Within a frictional western boundary current, the dominant balance is between advection of planetary
vorticity and the frictional vorticity sink:
∂v
βv ≈ −r
∂x
(where we have assumed |v| ≫ |u| in the boundary current).
Scaling the above, we find δ ∼ r/β as required (or one can solve for the exponential boundary layer directly).
r ∼ (spin down time-scale)−1 ∼ 0.3 × 10−7 s−1 . Thus
r 0.3 × 10−7 s−1
δ∼ ∼ = 3 × 103 m = 3 km.
β 2 × 10−11 m−1 s−1
Northward transport in the Sverdrup interior must be balanced by an opposite transport in the boundary
current, where T ∼ δHv.
Hence
T 3 × 107 m3 s−1
v∼ ∼ = 20 m s−1 ,
δH 3 × 103 m . 5 × 102 m
and
∂v 20 m s−1
ξ≈ ∼ ∼ 10−2 s−1 ≫ |f |
∂x 3 × 103 m
(let alone greater than variations in f ).
Hence relative vorticity is not negligible in western boundary currents.
[10]
[Total: 25]
11
5. The equations of motion for a two-dimensional ocean of uniform depth, forced by a zonal wind stress and
dissipated by linear friction, can be written:
∂u ∂u ∂u 1 ∂p τs
+u +v − fv + = − ru,
∂t ∂x ∂y ρ0 ∂x ρ0 H
∂v ∂v ∂v 1 ∂p
+u +v + fu + = −rv,
∂t ∂x ∂y ρ0 ∂y
∂u ∂v
+ = 0,
∂x ∂y
where the symbols have their usual meanings.
Derive the vorticity equation:
∂ ∂ ∂ 1 ∂τs
+u +v ζ =− − rξ,
∂t ∂x ∂y ρ0 H ∂y
where ζ is the absolute vorticity and ξ is the relative vorticity.
Taking the curl of the momentum equations gives:
∂ ∂ ∂ ∂u ∂v ∂u ∂v df 1 ∂τs
+u +v ξ+ξ + +f + +v =− − rξ,
∂t ∂x ∂y ∂x ∂y ∂x ∂y dy ρ0 H ∂y
where the relative vorticity is defined as
∂v ∂u
ξ= − ,
∂x ∂y
and we assume f = f (y) (although the result holds for arbitrary f ).
Using the continuity equation to eliminate two terms in the above, and defining the absolute vorticity
as ζ = ξ + f , the result follows:
∂ ∂ ∂ 1 ∂τs
+u +v ζ =− − rξ.
∂t ∂x ∂y ρ0 H ∂y
[5]
Write down the form of the vorticity equation known as “Sverdrup balance”, which holds approximately in
the interior of an ocean gyre. List the conditions required for its validity. Assuming that the wind stress
varies only with latitude, show that the northward volume transport predicted by Sverdrup balance is
L ∂τs
T =− ,
βρ0 ∂y
where L is the basin width and β has its usual meaning. Sketch the circulation predicted by Sverdrup balance
in a typical hemispheric ocean basin, stating clearly any boundary conditions that you assume. Estimate the
volume transport of a typical subtropical gyre, assuming a maximum wind stress of magnitude 0.2 N m−2 and
typical values for the remaining parameters.
The approximate Sverdrup balance is between advection of planetary vorticity and the source of vorticity
from the wind stress curl:
1 ∂τs
βv = − ,
ρ0 H ∂y
where β = df /dy.
This requires neglect of: relative vorticity, assumed small compared with variations in planetary vorticity; the
frictional vorticity sink, small outside the boundary currents; and time-dependence, negligible on time-scale
longer than the barotropic Rossby wave propagation time-scale. (There is no need to give mathematical
conditions for validity of these approximations.)
The northward transport predicted by Sverdrup balance is
L ∂τs
Z
Hv dx = − ,
βρ0 ∂y
as required.
9
,2009 C5: Physics of Atmospheres and Oceans
Assume easterly trades in the tropics, westerlies at mid-latitudes and easterlies (or at least weakening
westerlies) at high latitudes.
Also assume no flow through the eastern boundary, following Sverdrup’s original calculation, on the
basis that we know boundary currents form at the western margins of gyres where Sverdrup balance breaks
down.
Thus (please ignore Ψ labels):
polar
easterlies
subpolar
Ψ− ocean
mid−latitude
westerlies
subtropical
Ψ+ ocean
trades
τs
Gyre transport corresponds to maximum equatorward Sverdrup transport, as evident from the previous
figure.
Estimate wind stress curl as change in wind stress between equator and mid-latitudes (say 40◦ N):
∂τ 0.4 N m−2
∼ = 10−7 N m3 .
∂y 4 × 106
Take the width of the Atlantic (say) to be 5 000 km and β ≈ 2 × 10−11 m−1 s−1 and ρ0 ∼ 103 kg m−3 .
Thus
L ∂τs −1
Tgyre ≈ ∼ 2.5 × 107 m3 s (= 25 Sv).
βρ0 ∂y max
(Give full credit for any answer in the right ballpark here.)
[10]
10
, 2009 C5: Physics of Atmospheres and Oceans
Suppose that the gyre is closed by a frictional boundary current of width δ. By scaling the appropriate terms
in the vorticity equation, or otherwise, deduce that
r
δ∼ .
β
Assuming a frictional time scale of 1 year and H = 500 m, estimate δ and a typical boundary current velocity.
Comment on the extent to which these values are realistic. Using these values, estimate the magnitude of
the relative vorticity in the boundary current, and thus deduce that it cannot be neglected in reality.
Within a frictional western boundary current, the dominant balance is between advection of planetary
vorticity and the frictional vorticity sink:
∂v
βv ≈ −r
∂x
(where we have assumed |v| ≫ |u| in the boundary current).
Scaling the above, we find δ ∼ r/β as required (or one can solve for the exponential boundary layer directly).
r ∼ (spin down time-scale)−1 ∼ 0.3 × 10−7 s−1 . Thus
r 0.3 × 10−7 s−1
δ∼ ∼ = 3 × 103 m = 3 km.
β 2 × 10−11 m−1 s−1
Northward transport in the Sverdrup interior must be balanced by an opposite transport in the boundary
current, where T ∼ δHv.
Hence
T 3 × 107 m3 s−1
v∼ ∼ = 20 m s−1 ,
δH 3 × 103 m . 5 × 102 m
and
∂v 20 m s−1
ξ≈ ∼ ∼ 10−2 s−1 ≫ |f |
∂x 3 × 103 m
(let alone greater than variations in f ).
Hence relative vorticity is not negligible in western boundary currents.
[10]
[Total: 25]
11
