84
ANNE-MARIE TREGUIER
With constant κ this is a classical diffusion equation and the perturbation decays with a characteristic time τ = H 2 /(π 2 κ).
When κ varies vertically, the second term on the right-hand side of
(8) is non zero. It is similar to a vertical advection with velocity
w κ = −∂κ/∂z.
This term can lead to a sharpening of the large scale gradients (P. Klein,
personal communication). To see this, let us consider the equation for
the temperature gradient T z , obtained by taking the vertical derivative
of (8):
∂T z
∂t
= κ
∂ 2 T z
∂z 2 + 2
∂κ
∂z
∂T z
∂z
+
∂ 2 κ
∂z 2 T z .
(9)
The first term is the diffusion, the second the advective contribution,
and the third term can cause an exponential growth of the temperature
gradient when ∂ 2 κ/∂z 2 is large enough. This happens if κ varies more
rapidly in space than T . Let us assume, for instance, that the initial T
profile has some small scale variations superimposed on it, and that the
physical processes generating mixing are very sensitive to the presence
of those small scales. This situation is displayed in Fig. 5. The mixing
cofficient has large values where the small scales are present, in the upper third part of the water column. Instead of decaying, the profile of
temperature after 120 days has a much stronger gradient. The T profile, initially of typical scale H, varies now with the typical spatial scale
of the κ profile. The temperature perturbation has also been advected
downwards. Note that although a sharpening of the gradient has occurred, diffusion has smoothed the local extrema in the initial profile as
expected (this property of the diffusion equation is independent of the
structure of the diffusion coefficient).
Even more interesting things happen when κ is a nonlinear decreasing
function of ∂T /∂z. When κ is nonlinear enough, the parameterization
can generate discontinuities and staircases in the temperature profile.
Fig. 6 shows the final state of the evolution of eq.(8) with κ proportional
to exp(−(dT /dz) 2 ). This effect was noted by Phillips (1972) and more
recently by Ruddick et al. (1989)
Letting κ be a decreasing function of the vertical temperature gradient
is precisely what parametrizations of vertical mixing do: stratification
inhibit vertical mixing by providing a strong restoring force (buoyancy
force), thus limiting vertical displacements. Most parameterizations of
vertical mixing are based on the Richardson number of the large scale
flow. Let us define first the Vaisala frequency N :
N
2 =
−g∂ρ/∂z
ρ 0
,
ANNE-MARIE TREGUIER
With constant κ this is a classical diffusion equation and the perturbation decays with a characteristic time τ = H 2 /(π 2 κ).
When κ varies vertically, the second term on the right-hand side of
(8) is non zero. It is similar to a vertical advection with velocity
w κ = −∂κ/∂z.
This term can lead to a sharpening of the large scale gradients (P. Klein,
personal communication). To see this, let us consider the equation for
the temperature gradient T z , obtained by taking the vertical derivative
of (8):
∂T z
∂t
= κ
∂ 2 T z
∂z 2 + 2
∂κ
∂z
∂T z
∂z
+
∂ 2 κ
∂z 2 T z .
(9)
The first term is the diffusion, the second the advective contribution,
and the third term can cause an exponential growth of the temperature
gradient when ∂ 2 κ/∂z 2 is large enough. This happens if κ varies more
rapidly in space than T . Let us assume, for instance, that the initial T
profile has some small scale variations superimposed on it, and that the
physical processes generating mixing are very sensitive to the presence
of those small scales. This situation is displayed in Fig. 5. The mixing
cofficient has large values where the small scales are present, in the upper third part of the water column. Instead of decaying, the profile of
temperature after 120 days has a much stronger gradient. The T profile, initially of typical scale H, varies now with the typical spatial scale
of the κ profile. The temperature perturbation has also been advected
downwards. Note that although a sharpening of the gradient has occurred, diffusion has smoothed the local extrema in the initial profile as
expected (this property of the diffusion equation is independent of the
structure of the diffusion coefficient).
Even more interesting things happen when κ is a nonlinear decreasing
function of ∂T /∂z. When κ is nonlinear enough, the parameterization
can generate discontinuities and staircases in the temperature profile.
Fig. 6 shows the final state of the evolution of eq.(8) with κ proportional
to exp(−(dT /dz) 2 ). This effect was noted by Phillips (1972) and more
recently by Ruddick et al. (1989)
Letting κ be a decreasing function of the vertical temperature gradient
is precisely what parametrizations of vertical mixing do: stratification
inhibit vertical mixing by providing a strong restoring force (buoyancy
force), thus limiting vertical displacements. Most parameterizations of
vertical mixing are based on the Richardson number of the large scale
flow. Let us define first the Vaisala frequency N :
N
2 =
−g∂ρ/∂z
ρ 0
,