Warming of the oceans by the atmosphere

Author

Roland Van den Broek - Civil engineer

Date de publication

1 septembre 2024

1 Introduction

To what extent can an atmosphere warmer than the ocean warm it?

The following calculations give an order of magnitude of this warming.

We consider a column of atmosphere of $1 M^{2}$ with a height $H_{a}$ in contact with an ocean column of the same cross-section and height $H_{o}$ .

The height $H_{a}$ taken into account is equal to the total mass of the atmosphere located above the section of $1 M^{2}$ . It is given by the following calculation:

$\begin{aligned} P_{a t m} & = T h e a t m o s p h e r i c p r e s s u r e \\ = 101325 M P a \\ g & = T h e g r a v i t y a c c e l e r a t i o n \\ = 9.81 m / s^{2} \\ ρ_{a i r} & = 1.293 k g / M^{3} \\ H_{a} & = P_{a t m} / g / ρ_{a i r} \\ = 7988 M \end{aligned}$

We used the specific heat capacities $C_{a} = 1.256 k J / M 3 / ° K$ (atmosphere), $C_{o} = 4200 k J / M 3 / K$ (ocean).

We assumed that the air and ocean columns have a homogeneous initial temperature $T_{a}$ (air) and $T_{o}$ (ocean).

The equilibrium temperature of the air and ocean $T_{e q}$ is given by the following formula:

$T_{e q} = \frac{T_{a} \cdot C_{a} \cdot V_{a} + T_{o} \cdot C_{o} \cdot V_{o}}{C_{a} \cdot V_{a} + C_{o} \cdot V_{o}}$

Different values of $H_{o}$ , $T_{a}$ , and $T_{o}$ were used in the calculations in the following sections.

It is clear that it is extremely difficult to heat water with warm air.

For reference, the average depth of the oceans is 3,682 meters.

Note: The 33°C temperature difference is the one that NASA estimates is due to the greenhouse effect.

2 Example: $T_{a} - T_{o} =$ 1 $° C$ , $H_{o} =$ 20 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 20 $M^{3}$	Ocean volume
$H_{o} =$ 20 $M$	Ocean height
$T_{a} =$ 16 $° C$	Initial air temperature
$T_{o} =$ 15 $° C$	Initial ocean temperature
$T_{e q} =$ 15.1067 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.1067 $° C$	Ocean warming

3 Example: $T_{a} - T_{o} =$ 1 $° C$ , $H_{o} =$ 50 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 50 $M^{3}$	Ocean volume
$H_{o} =$ 50 $M$	Ocean height
$T_{a} =$ 16 $° C$	Initial air temperature
$T_{o} =$ 15 $° C$	Initial ocean temperature
$T_{e q} =$ 15.0456 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.0456 $° C$	Ocean warming

4 Example: $T_{a} - T_{o} =$ 1 $° C$ , $H_{o} =$ 100 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 100 $M^{3}$	Ocean volume
$H_{o} =$ 100 $M$	Ocean height
$T_{a} =$ 16 $° C$	Initial air temperature
$T_{o} =$ 15 $° C$	Initial ocean temperature
$T_{e q} =$ 15.0233 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.0233 $° C$	Ocean warming

5 Example: $T_{a} - T_{o} =$ 1 $° C$ , $H_{o} =$ 1000 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 1000 $M^{3}$	Ocean volume
$H_{o} =$ 1000 $M$	Ocean height
$T_{a} =$ 16 $° C$	Initial air temperature
$T_{o} =$ 15 $° C$	Initial ocean temperature
$T_{e q} =$ 15.0024 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.0024 $° C$	Ocean warming

6 Example: $T_{a} - T_{o} =$ 1 $° C$ , $H_{o} =$ 3382 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 3382 $M^{3}$	Ocean volume
$H_{o} =$ 3382 $M$	Ocean height
$T_{a} =$ 16 $° C$	Initial air temperature
$T_{o} =$ 15 $° C$	Initial ocean temperature
$T_{e q} =$ 15.0007 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 7^{-4} $° C$	Ocean warming

7 Example: $T_{a} - T_{o} =$ 33 $° C$ , $H_{o} =$ 20 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 20 $M^{3}$	Ocean volume
$H_{o} =$ 20 $M$	Ocean height
$T_{a} =$ 15 $° C$	Initial air temperature
$T_{o} =$ -18 $° C$	Initial ocean temperature
$T_{e q} =$ -14.479 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 3.521 $° C$	Ocean warming

8 Example: $T_{a} - T_{o} =$ 33 $° C$ , $H_{o} =$ 50 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 50 $M^{3}$	Ocean volume
$H_{o} =$ 50 $M$	Ocean height
$T_{a} =$ 15 $° C$	Initial air temperature
$T_{o} =$ -18 $° C$	Initial ocean temperature
$T_{e q} =$ -16.4953 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 1.5047 $° C$	Ocean warming

9 Example: $T_{a} - T_{o} =$ 33 $° C$ , $H_{o} =$ 100 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 100 $M^{3}$	Ocean volume
$H_{o} =$ 100 $M$	Ocean height
$T_{a} =$ 15 $° C$	Initial air temperature
$T_{o} =$ -18 $° C$	Initial ocean temperature
$T_{e q} =$ -17.2301 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.7699 $° C$	Ocean warming

10 Example: $T_{a} - T_{o} =$ 33 $° C$ , $H_{o} =$ 1000 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 1000 $M^{3}$	Ocean volume
$H_{o} =$ 1000 $M$	Ocean height
$T_{a} =$ 15 $° C$	Initial air temperature
$T_{o} =$ -18 $° C$	Initial ocean temperature
$T_{e q} =$ -17.9214 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.0786 $° C$	Ocean warming

11 Example: $T_{a} - T_{o} =$ 33 $° C$ , $H_{o} =$ 3382 $M$

$V_{a} =$ 7988 $M^{3}$	Air volume at 1 atm
$H_{a} =$ 7988 $M$	Height
$C_{a} =$ 1256 $J / M^{3} / ° K$	Volumetric heat capacity of air
$C_{o} =$ 4200000 $J / M^{3} / ° K$	Ocean volumetric heat capacity
$V_{o} =$ 3382 $M^{3}$	Ocean volume
$H_{o} =$ 3382 $M$	Ocean height
$T_{a} =$ 15 $° C$	Initial air temperature
$T_{o} =$ -18 $° C$	Initial ocean temperature
$T_{e q} =$ -17.9767 $° C$	Equilibrium temperature of air and ocean
$Δ T_{o} =$ 0.0233 $° C$	Ocean warming

1 Introduction

2 Example: Ta−To= 1 °C, Ho= 20 M

3 Example: Ta−To= 1 °C, Ho= 50 M

4 Example: Ta−To= 1 °C, Ho= 100 M

5 Example: Ta−To= 1 °C, Ho= 1000 M

6 Example: Ta−To= 1 °C, Ho= 3382 M

7 Example: Ta−To= 33 °C, Ho= 20 M

8 Example: Ta−To= 33 °C, Ho= 50 M

9 Example: Ta−To= 33 °C, Ho= 100 M

10 Example: Ta−To= 33 °C, Ho= 1000 M

11 Example: Ta−To= 33 °C, Ho= 3382 M