2010 C5: Physics of Atmospheres and Oceans
6. Give a brief account (with examples in the Earth’s oceans and atmosphere) of what is meant by the term
western intensification and its physical origin.
Western Intensification: tendency of a poleward flow to become stronger and concentrated towards a western
boundary. Occurs essentially when the main geostrophic component of the flow is unable to satisfy lateral [?]
boundary conditions, e.g., in the Earth’s oceans, geostrophic flow driven by surface wind stress so vorticity
balance is between poleward advection and wind stress curl.
βv ≈ ∇ × τ , sense of v determined by β. Satisfies boundary conditions at eastern boundary but not at
western boundary, hence frictional or inertial effects become important. Examples: Gulf Stream, Kuroshio,
Agulhas currents in the ocean, Somali jet in the Earth’s atmosphere.
[4]
The Tharsis Plateau is a major topographic ridge on Mars, oriented north-south across the equator and
blocking the low level zonal flow between latitudes 30◦ N and 30◦ S. The low-level flow at these latitudes can
be assumed to be barotropic and non-divergent on an equatorial β-plane, and approximately satisfies
∂Φ
−βyv = − − r[u − u0 (y)], (1)
∂x
∂Φ
βyu = − − rv, (2)
∂y
∂u ∂v
+ = 0. (3)
∂x ∂y
Here Φ is geopotential height and β has its usual meteorological definition. The function u0 (y) represents the
pattern of net forcing of the zonal flow by seasonal differential heating and r is a constant. Give a physical
interpretation of these equations and account for the approximations which lead to them. Show that they
reduce to the single equation βv + r∇2 ψ = F (y) and give appropriate definitions of the functions F (y) and
ψ in terms of u, v and u0 .
Eqs (1) and (2) are horizontal momentum equations in steady-state on an equatorial β-plane in pressure
coordinates ( Φ is geopotential of pressure surfaces). Friction approximated by a linear drag with Rayleigh
coefficient r. Same timescale assumed to apply to forcing in Eq (1). Eq. (3) is the continuity equation for a
barotropic flow. [3]
Forming a vorticity equation by taking the curl of the momentum equations: ∂/∂x(2) − ∂/∂y(1):
∂u ∂v ∂v ∂u du0
βv + βy + = −r − −r
∂x ∂y =0 ∂x ∂y dy
Putting u, v in terms of a stream function ψ: u = −∂ψ/∂y, v = ∂ψ/∂x, (∂v/∂x − ∂u/∂y) = ∇2 ψ,
∂ψ du0
β + r∇2 ψ = −r ≡ F (y) [3]
∂x dy
[6]
The low-level atmospheric circulation during the southern hemisphere winter is characterized by westward
zonal flow at northern mid-latitudes and eastward flow at southern mid-latitudes, which can be represented
by u0 (y) = −U0 sin(πy/L) (where U0 is a constant) with the equator at y = 0 and the eastern boundary of
the Tharsis Plateau at x = 0 between −L/2 ≤ y ≤ L/2. Obtain forced solutions of the form ψ = A(x)B(y)
with suitable boundary conditions in x and y. Sketch the streamlines, showing the direction of flow.
Setting u0 (y) = −U0 sin(πy/L),
du0 rπU0 πy
F (y) ≡ −r = cos
dy L L
∂ψ rπU0 πy
β = −r∇2 ψ + cos
∂x L L
Try a solution ψ = A(x) cos(πy/L):
d2 A π2
dA rπU0
β = −r − A +
dx dx2 L2 L
which can be rearranged as
d2 A β dA π2 πU0
2
+ − 2A =
dx r dx L L
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6. Give a brief account (with examples in the Earth’s oceans and atmosphere) of what is meant by the term
western intensification and its physical origin.
Western Intensification: tendency of a poleward flow to become stronger and concentrated towards a western
boundary. Occurs essentially when the main geostrophic component of the flow is unable to satisfy lateral [?]
boundary conditions, e.g., in the Earth’s oceans, geostrophic flow driven by surface wind stress so vorticity
balance is between poleward advection and wind stress curl.
βv ≈ ∇ × τ , sense of v determined by β. Satisfies boundary conditions at eastern boundary but not at
western boundary, hence frictional or inertial effects become important. Examples: Gulf Stream, Kuroshio,
Agulhas currents in the ocean, Somali jet in the Earth’s atmosphere.
[4]
The Tharsis Plateau is a major topographic ridge on Mars, oriented north-south across the equator and
blocking the low level zonal flow between latitudes 30◦ N and 30◦ S. The low-level flow at these latitudes can
be assumed to be barotropic and non-divergent on an equatorial β-plane, and approximately satisfies
∂Φ
−βyv = − − r[u − u0 (y)], (1)
∂x
∂Φ
βyu = − − rv, (2)
∂y
∂u ∂v
+ = 0. (3)
∂x ∂y
Here Φ is geopotential height and β has its usual meteorological definition. The function u0 (y) represents the
pattern of net forcing of the zonal flow by seasonal differential heating and r is a constant. Give a physical
interpretation of these equations and account for the approximations which lead to them. Show that they
reduce to the single equation βv + r∇2 ψ = F (y) and give appropriate definitions of the functions F (y) and
ψ in terms of u, v and u0 .
Eqs (1) and (2) are horizontal momentum equations in steady-state on an equatorial β-plane in pressure
coordinates ( Φ is geopotential of pressure surfaces). Friction approximated by a linear drag with Rayleigh
coefficient r. Same timescale assumed to apply to forcing in Eq (1). Eq. (3) is the continuity equation for a
barotropic flow. [3]
Forming a vorticity equation by taking the curl of the momentum equations: ∂/∂x(2) − ∂/∂y(1):
∂u ∂v ∂v ∂u du0
βv + βy + = −r − −r
∂x ∂y =0 ∂x ∂y dy
Putting u, v in terms of a stream function ψ: u = −∂ψ/∂y, v = ∂ψ/∂x, (∂v/∂x − ∂u/∂y) = ∇2 ψ,
∂ψ du0
β + r∇2 ψ = −r ≡ F (y) [3]
∂x dy
[6]
The low-level atmospheric circulation during the southern hemisphere winter is characterized by westward
zonal flow at northern mid-latitudes and eastward flow at southern mid-latitudes, which can be represented
by u0 (y) = −U0 sin(πy/L) (where U0 is a constant) with the equator at y = 0 and the eastern boundary of
the Tharsis Plateau at x = 0 between −L/2 ≤ y ≤ L/2. Obtain forced solutions of the form ψ = A(x)B(y)
with suitable boundary conditions in x and y. Sketch the streamlines, showing the direction of flow.
Setting u0 (y) = −U0 sin(πy/L),
du0 rπU0 πy
F (y) ≡ −r = cos
dy L L
∂ψ rπU0 πy
β = −r∇2 ψ + cos
∂x L L
Try a solution ψ = A(x) cos(πy/L):
d2 A π2
dA rπU0
β = −r − A +
dx dx2 L2 L
which can be rearranged as
d2 A β dA π2 πU0
2
+ − 2A =
dx r dx L L
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