stommel's ocean circulation box model
TRANSCRIPT
![Page 1: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/1.jpg)
Stommel’s Ocean Circulation Box Model
Julie Leifeld
University of Minnesota
April 2, 2012
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 2: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/2.jpg)
The Premise
I Ocean circulation is driven by density differences (amongother things)
I Density is affected by temperature and salinity
I Temperature and salinity work in opposing ways. Which effectdominates circulation?
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 3: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/3.jpg)
The Premise
I Ocean circulation is driven by density differences (amongother things)
I Density is affected by temperature and salinity
I Temperature and salinity work in opposing ways. Which effectdominates circulation?
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 4: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/4.jpg)
The Premise
I Ocean circulation is driven by density differences (amongother things)
I Density is affected by temperature and salinity
I Temperature and salinity work in opposing ways. Which effectdominates circulation?
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 5: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/5.jpg)
The First Model
dTdt = c(T − T )dSdt = d(S − S)
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 6: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/6.jpg)
The First Model
First: Nondimensionalize the equations.
τ = ct, δ =d
c, y =
T
T, x =
S
S
The new equations:
dydτ = 1− ydxdτ = δ(1− x)
δ is considered to be small. This means that salinity changes areslower than temperature changes.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 7: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/7.jpg)
The First Model
First: Nondimensionalize the equations.
τ = ct, δ =d
c, y =
T
T, x =
S
S
The new equations:
dydτ = 1− ydxdτ = δ(1− x)
δ is considered to be small. This means that salinity changes areslower than temperature changes.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 8: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/8.jpg)
The First Model
First: Nondimensionalize the equations.
τ = ct, δ =d
c, y =
T
T, x =
S
S
The new equations:
dydτ = 1− ydxdτ = δ(1− x)
δ is considered to be small. This means that salinity changes areslower than temperature changes.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 9: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/9.jpg)
The First Model
First: Nondimensionalize the equations.
τ = ct, δ =d
c, y =
T
T, x =
S
S
The new equations:
dydτ = 1− ydxdτ = δ(1− x)
δ is considered to be small. This means that salinity changes areslower than temperature changes.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 10: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/10.jpg)
The First Model
First: Nondimensionalize the equations.
τ = ct, δ =d
c, y =
T
T, x =
S
S
The new equations:
dydτ = 1− ydxdτ = δ(1− x)
δ is considered to be small. This means that salinity changes areslower than temperature changes.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 11: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/11.jpg)
The First Model
These are both 1D linear equations. We can solve them!
Thesolution is:
y = 1 + (y0 − 1)e−τ
x = 1 + (x0 − 1)e−δτ
Notice: x , y → 1, but y → 1 more quickly. Recall: x = 1 = y ⇒x = S, y = T .
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 12: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/12.jpg)
The First Model
These are both 1D linear equations. We can solve them! Thesolution is:
y = 1 + (y0 − 1)e−τ
x = 1 + (x0 − 1)e−δτ
Notice: x , y → 1, but y → 1 more quickly. Recall: x = 1 = y ⇒x = S, y = T .
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 13: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/13.jpg)
The First Model
These are both 1D linear equations. We can solve them! Thesolution is:
y = 1 + (y0 − 1)e−τ
x = 1 + (x0 − 1)e−δτ
Notice: x , y → 1, but y → 1 more quickly. Recall: x = 1 = y ⇒x = S, y = T .
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 14: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/14.jpg)
The First Model
These are both 1D linear equations. We can solve them! Thesolution is:
y = 1 + (y0 − 1)e−τ
x = 1 + (x0 − 1)e−δτ
Notice: x , y → 1, but y → 1 more quickly. Recall: x = 1 = y ⇒x = S, y = T .
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 15: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/15.jpg)
The First Model
How is density affected by Temperature and Salinity?
ρ = ρ0(1− αT + βS)
In terms of x and y ,
ρ = ρ0(1 + αT (−y + Rx)
R = βSαT R measures the effect of salinity vs. temperature at the
final equilibrium, x = y = 1.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 16: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/16.jpg)
The First Model
How is density affected by Temperature and Salinity?
ρ = ρ0(1− αT + βS)
In terms of x and y ,
ρ = ρ0(1 + αT (−y + Rx)
R = βSαT R measures the effect of salinity vs. temperature at the
final equilibrium, x = y = 1.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 17: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/17.jpg)
The First Model
How is density affected by Temperature and Salinity?
ρ = ρ0(1− αT + βS)
In terms of x and y ,
ρ = ρ0(1 + αT (−y + Rx)
R = βSαT
R measures the effect of salinity vs. temperature at thefinal equilibrium, x = y = 1.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 18: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/18.jpg)
The First Model
How is density affected by Temperature and Salinity?
ρ = ρ0(1− αT + βS)
In terms of x and y ,
ρ = ρ0(1 + αT (−y + Rx)
R = βSαT R measures the effect of salinity vs. temperature at the
final equilibrium, x = y = 1.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 19: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/19.jpg)
The First Model
How does the density change over time?
dρ
dτ= ρ0(αT )(−dy
dτ+ R
dx
dτ) = ρ0(αT )(y − 1 + Rδ(1− x))
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 20: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/20.jpg)
The First Model
How does the density change over time?
dρ
dτ= ρ0(αT )(−dy
dτ+ R
dx
dτ) = ρ0(αT )(y − 1 + Rδ(1− x))
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 21: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/21.jpg)
The First Model
Consider the case: Rδ < 1, R > 1, x0 = y0 = 0
When τ = 0,dρdτ = ρ0(αT )(−1 + Rδ) < 0 Hidden assumption: T > 0 Whenτ →∞, ρ = ρ0(1 + αT (−1 + R)) > ρ0 The point: Temperature isthe dominant effect at first, but after sufficient time, salinity effectstake over, and the final density is higher than the initial density.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 22: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/22.jpg)
The First Model
Consider the case: Rδ < 1, R > 1, x0 = y0 = 0 When τ = 0,dρdτ = ρ0(αT )(−1 + Rδ) < 0 Hidden assumption: T > 0
Whenτ →∞, ρ = ρ0(1 + αT (−1 + R)) > ρ0 The point: Temperature isthe dominant effect at first, but after sufficient time, salinity effectstake over, and the final density is higher than the initial density.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 23: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/23.jpg)
The First Model
Consider the case: Rδ < 1, R > 1, x0 = y0 = 0 When τ = 0,dρdτ = ρ0(αT )(−1 + Rδ) < 0 Hidden assumption: T > 0 Whenτ →∞, ρ = ρ0(1 + αT (−1 + R)) > ρ0
The point: Temperature isthe dominant effect at first, but after sufficient time, salinity effectstake over, and the final density is higher than the initial density.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 24: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/24.jpg)
The First Model
Consider the case: Rδ < 1, R > 1, x0 = y0 = 0 When τ = 0,dρdτ = ρ0(αT )(−1 + Rδ) < 0 Hidden assumption: T > 0 Whenτ →∞, ρ = ρ0(1 + αT (−1 + R)) > ρ0 The point: Temperature isthe dominant effect at first, but after sufficient time, salinity effectstake over, and the final density is higher than the initial density.
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 25: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/25.jpg)
The First Model
Consider the density anomaly: σ =
“ρρ0−1
”αT
Until τ = 1, ρ isdecreasing (the densityanomaly is getting morenegative) After τ = 1, ρ isincreasing. After τ = 4,ρ ≥ ρ0
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 26: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/26.jpg)
The First Model
Consider the density anomaly: σ =
“ρρ0−1
”αT
Until τ = 1, ρ isdecreasing (the densityanomaly is getting morenegative) After τ = 1, ρ isincreasing. After τ = 4,ρ ≥ ρ0
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 27: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/27.jpg)
The First Model
Consider the density anomaly: σ =
“ρρ0−1
”αT
Until τ = 1, ρ isdecreasing (the densityanomaly is getting morenegative)
After τ = 1, ρ isincreasing. After τ = 4,ρ ≥ ρ0
Julie Leifeld Stommel’s Ocean Circulation Box Model
![Page 28: Stommel's Ocean Circulation Box Model](https://reader033.vdocuments.mx/reader033/viewer/2022061101/629bf4aa23976570706097f3/html5/thumbnails/28.jpg)
The First Model
Consider the density anomaly: σ =
“ρρ0−1
”αT
Until τ = 1, ρ isdecreasing (the densityanomaly is getting morenegative) After τ = 1, ρ isincreasing. After τ = 4,ρ ≥ ρ0
Julie Leifeld Stommel’s Ocean Circulation Box Model