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Surface model

Storyboard

The transfer of particles or molecules, such as CO2, between the atmosphere and the ocean involves a more complex mechanism. This process is associated with the formation of a liquid film saturated with particles or molecules, which regulates the movement of new particles to or from the interior of the ocean.

>Model

ID:(1633, 0)



Mechanisms

Iframe

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Code
Concept
CO2 exchange, speed from water
Solubility as a function of Schmidt number
Surface layer
Transfer speed
Transfer speed and resistances

Mechanisms

CO2 exchange, speed from waterSolubility as a function of Schmidt numberSurface layerTransfer speedTransfer speed and resistances

ID:(15640, 0)



Surface layer

Description

>Top


To study the transfer of CO2 at the ocean surface, it is necessary to carefully observe the concentration changes in both the air and the water.

In the air, CO2 enters the water, creating a zone of low concentration where it decreases from C_a to C_{a,0}. This layer has a thickness ranging from 0.1 to 1 mm.

The CO2 that enters the water initially accumulates at the surface, creating a concentration C_{w,0}, which then diffuses into the interior, reaching a lower concentration of C_w.

The concentration reduction allows us to define two zones: a very thin one, ranging from 0.02 to 0.2 mm, where the concentration decreases rapidly, and a second zone, spanning from 0.6 to 2 mm, where the concentration decreases more gradually until it reaches the concentration within the water.

Ocean-Atmosphere Interactions of Gases and Particles, Peter S. Liss, Martin T. Johnson (eds.), Springer-Verlag Berlin Heidelberg

ID:(12244, 0)



CO2 exchange, speed from water

Concept

>Top


The gas transfer rate in water (k_w) can be modeled using measured data. Firstly, it depends on the rate at which the system removes carbon from the air-water interface, making the transport speed proportional to the relative velocity between the two mediums.

Secondly, there is an effect of ion mobility, which can be described by the schmidt number (Sc), representing the relationship between momentum diffusion and particles. However, this dependence is not linear and is influenced by a factor exponente de Schmidt (n) that varies between -1/2 and -2/3 depending on the surface roughness.

Finally, the gas transfer rate in water (k_w) also depends on the factor beta del transporte aire a agua de CO2 (\beta), which in turn is determined by the level of surface roughness.

In summary, the gas the gas transfer rate in water (k_w) is described as a function of the velocidad del agua (u_w), the velocidad del aire (u_a), the schmidt number (Sc), the factor beta del transporte aire a agua de CO2 (\beta), and exponente de Schmidt (n) as follows:

k_w = ( u_a - u_w ) \beta Sc ^ n

ID:(15652, 0)



Solubility as a function of Schmidt number

Concept

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The mobility of molecules, represented by the gas solubility (\alpha), is modeled as a function of particle concentration, characterized by the schmidt number (Sc), which in turn is calculated from parameters the viscosidad en masa acuosa (\eta), the densidad en capa de masa acuosa (\rho), and the constante de difusión en masa acuosa (D) using the following expression:

Sc =\displaystyle\frac{ \eta }{ \rho D }



This relationship is visualized in the following diagram:

Ocean-Atmosphere Interactions of Gases and Particles, Peter S. Liss, Martin T. Johnson (eds.), Springer-Verlag Berlin Heidelberg

ID:(12245, 0)



Transfer speed

Concept

>Top


The gas transfer rate in air (k_a) can be estimated from Fick's law by comparing the constante de difusión en masa acuosa (D) with the grosor de la capa superficial (\delta_c) as follows:

k_a = \displaystyle\frac{ D }{ \delta_c }

ID:(15653, 0)



Transfer speed and resistances

Concept

>Top


For the interaction between the atmosphere and the ocean, the air to water transfer resistance of a gas (R_{ta}) initially encompasses the transfer resistance in water (R_w), followed by the evaporation process 1/\alpha with the gas solubility (\alpha), and once the gas has passed into the air, the transfer resistance in air (R_a) acts upon it:

R_{ta} = R_a + \displaystyle\frac{1}{ \alpha } R_w



Meanwhile, in the interaction between the atmosphere and the ocean, the water to air transfer resistance of a gas (R_{tw}) initially includes the transfer resistance in air (R_a), followed by the gas solubility (\alpha), and once the gas has penetrated the water, the transfer resistance in water (R_w) comes into play:

R_{tw} = R_w + \alpha R_a



With these equations, we can formulate the relationships for the transfer velocities.

Therefore, using the total transfer rate of gas in air (k_{ta}), the gas transfer rate in water (k_w), the gas transfer rate in air (k_a), and the gas solubility (\alpha), we establish the following relationship:

\displaystyle\frac{1}{ k_{ta} } = \displaystyle\frac{1}{ k_a } + \displaystyle\frac{1}{ \alpha k_w }



On the other hand, with the total transfer rate of gas in water (k_{tw}), the gas transfer rate in water (k_w), the gas transfer rate in air (k_a), and the gas solubility (\alpha), we establish that:

\displaystyle\frac{1}{ k_{tw} } = \displaystyle\frac{1}{ k_w } + \displaystyle\frac{ \alpha }{ k_a }

ID:(15654, 0)



Model

Top

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Parameters

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
D
D
Constante de difusión en masa acuosa
m/s^2
n
n
Exponente de Schmidt
-
\beta
beta
Factor beta del transporte aire a agua de CO2
-
\alpha
alpha
Gas solubility
-
k_a
k_a
Gas transfer rate in air
m/s
\delta_c
delta_c
Grosor de la capa superficial
m
Sc
Sc
Schmidt number
-
u_w
u_w
Velocidad del agua
m/s
u_a
u_a
Velocidad del aire
m/s

Variables

Symbol
Text
Variable
Value
Units
Calculate
MKS Value
MKS Units
R_{ta}
R_ta
Air to water transfer resistance of a gas
s/m
C_a
C_a
Gas concentration in the atmosphere
1/m^3
C_{w,0}
C_w0
Gas concentration in water
1/m^3
k_w
k_w
Gas transfer rate in water
m/s
k_{ta}
k_ta
Total transfer rate of gas in air
m/s
k_{tw}
k_tw
Total transfer rate of gas in water
m/s
R_a
R_a
Transfer resistance in air
s/m
R_w
R_w
Transfer resistance in water
s/m
R_{tw}
R_tw
Water to air transfer resistance of a gas
s/m

Calculations


First, select the equation: to , then, select the variable: to
1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a C_w = alpha * C_a k_a = D / delta_c k_w = ( u_a - u_w )* beta * Sc ^ n R_a = 1/ k_a R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = 1/ k_tw R_tw = R_w + alpha * R_a R_w = 1/ k_w R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

Calculations

Symbol
Equation
Solved
Translated

Calculations

Symbol
Equation
Solved
Translated

Variable Given Calculate Target : Equation To be used
1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a C_w = alpha * C_a k_a = D / delta_c k_w = ( u_a - u_w )* beta * Sc ^ n R_a = 1/ k_a R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = 1/ k_tw R_tw = R_w + alpha * R_a R_w = 1/ k_w R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw




Equations

#
Equation

\displaystyle\frac{1}{ k_{ta} } = \displaystyle\frac{1}{ k_a } + \displaystyle\frac{1}{ \alpha k_w }

1/ k_ta = 1/ k_a + 1/( k_w * alpha )


\displaystyle\frac{1}{ k_{tw} } = \displaystyle\frac{1}{ k_w } + \displaystyle\frac{ \alpha }{ k_a }

1/ k_tw = 1/ k_w + alpha / k_a


C_w = \alpha C_a

C_w = alpha * C_a


k_a = \displaystyle\frac{ D }{ \delta_c }

k_a = D / delta_c


k_w = ( u_a - u_w ) \beta Sc ^ n

k_w = ( u_a - u_w )* beta * Sc ^ n


R_a = \displaystyle\frac{1}{ k_a }

R_a = 1/ k_a


R_{ta} = \displaystyle\frac{1}{ k_{ta} }

R_ta = 1/ k_ta


R_{ta} = R_a + \displaystyle\frac{1}{ \alpha } R_w

R_ta = R_a + R_w / alpha


R_{tw} = \displaystyle\frac{1}{ k_{tw} }

R_tw = 1/ k_tw


R_{tw} = R_w + \alpha R_a

R_tw = R_w + alpha * R_a


R_w = \displaystyle\frac{1}{ k_w }

R_w = 1/ k_w

ID:(15645, 0)



Surface Concentration Relationships

Equation

>Top, >Model


The concentration gradient between the gas concentration in the atmosphere (C_a) and the gas concentration in water (C_{w,0}) depends on the gas solubility (\alpha). Therefore, the following relationship is established:

C_w = \alpha C_a

C_a
Gas concentration in the atmosphere
1/m^3
9416
C_w
Gas concentration in water
1/m^3
9415
\alpha
Gas solubility
-
9406
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12235, 0)



Transfer speed

Equation

>Top, >Model


The gas transfer rate in air (k_a) can be estimated from Fick's law by comparing the constante de difusión en masa acuosa (D) with the grosor de la capa superficial (\delta_c) as follows:

k_a = \displaystyle\frac{ D }{ \delta_c }

D
Constante de difusión en masa acuosa
m^2/s
9414
k_a
Gas transfer rate in air
m/s
9407
\delta_c
Grosor de la capa superficial
m
9430
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12227, 0)



CO2 exchange, speed from water

Equation

>Top, >Model


The gas parameter the gas transfer rate in water (k_w) is described in terms of the velocidad del agua (u_w), the velocidad del aire (u_a), the schmidt number (Sc), the factor beta del transporte aire a agua de CO2 (\beta), and exponente de Schmidt (n) as follows:

k_w = ( u_a - u_w ) \beta Sc ^ n

n
Exponente de Schmidt
-
9926
\beta
Factor beta del transporte aire a agua de CO2
-
9409
k_w
Gas transfer rate in water
m/s
9405
Sc
Schmidt number
-
9410
u_w
Velocidad del agua
m/s
9437
u_a
Velocidad del aire
m/s
9408
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12215, 0)



Transfer rate and resistance in the atmosphere

Equation

>Top, >Model


The transfer resistance in air (R_a) is defined as the inverse of the gas transfer rate in air (k_a). This relationship can be expressed as follows:

R_a = \displaystyle\frac{1}{ k_a }

k_a
Gas transfer rate in air
m/s
9407
R_a
Transfer resistance in air
s/m
9443
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12236, 0)



Transfer rate and resistance in the ocean

Equation

>Top, >Model


The transfer resistance in water (R_w) is defined as the inverse of the gas transfer rate in water (k_w). This relationship can be expressed as follows:

R_w = \displaystyle\frac{1}{ k_w }

k_w
Gas transfer rate in water
m/s
9405
R_w
Transfer resistance in water
s/m
9444
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12237, 0)



Total atmosphere-ocean transfer rate

Equation

>Top, >Model


The water to air transfer resistance of a gas (R_{tw}) is defined as the inverse of the total transfer rate of gas in water (k_{tw}). This relationship can be expressed as follows:

R_{tw} = \displaystyle\frac{1}{ k_{tw} }

k_{tw}
Total transfer rate of gas in water
m/s
9448
R_{tw}
Water to air transfer resistance of a gas
s/m
9446
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12238, 0)



Total ocean-atmosphere transfer rate

Equation

>Top, >Model


The air to water transfer resistance of a gas (R_{ta}) is defined as the inverse of the total transfer rate of gas in air (k_{ta}). This relationship can be expressed as follows:

R_{ta} = \displaystyle\frac{1}{ k_{ta} }

R_{ta}
Air to water transfer resistance of a gas
s/m
9445
k_{ta}
Total transfer rate of gas in air
m/s
9447
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12239, 0)



Total resistance to ocean-atmosphere transfer

Equation

>Top, >Model


For the interaction between the atmosphere and the ocean, the air to water transfer resistance of a gas (R_{ta}) initially includes the transfer resistance in water (R_w), followed by the evaporation process 1/\alpha with the gas solubility (\alpha). Once the gas has transferred to the air, the transfer resistance in air (R_a) acts on it:

R_{ta} = R_a + \displaystyle\frac{1}{ \alpha } R_w

R_{ta}
Air to water transfer resistance of a gas
s/m
9445
\alpha
Gas solubility
-
9406
R_a
Transfer resistance in air
s/m
9443
R_w
Transfer resistance in water
s/m
9444
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12240, 0)



Total resistance to atmosphere-ocean transfer

Equation

>Top, >Model


For the interaction between the atmosphere and the ocean, the water to air transfer resistance of a gas (R_{tw}) initially includes the transfer resistance in air (R_a), followed by the gas solubility (\alpha), and once the gas has penetrated the water, the transfer resistance in water (R_w) acts:

R_{tw} = R_w + \alpha R_a

\alpha
Gas solubility
-
9406
R_a
Transfer resistance in air
s/m
9443
R_w
Transfer resistance in water
s/m
9444
R_{tw}
Water to air transfer resistance of a gas
s/m
9446
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

ID:(12241, 0)



Atmosphere-Ocean Transfer Rate

Equation

>Top, >Model


The relationship between the transfer resistance between the atmosphere and the ocean can be expressed in terms of the transfer speeds in both mediums, equating to the inverse of the total transfer speed.

Therefore, with the total transfer rate of gas in water (k_{tw}), the gas transfer rate in water (k_w), the gas transfer rate in air (k_a), and the gas solubility (\alpha), it is established that:

\displaystyle\frac{1}{ k_{tw} } = \displaystyle\frac{1}{ k_w } + \displaystyle\frac{ \alpha }{ k_a }

\alpha
Gas solubility
-
9406
k_a
Gas transfer rate in air
m/s
9407
k_w
Gas transfer rate in water
m/s
9405
k_{tw}
Total transfer rate of gas in water
m/s
9448
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

The relationship between the water to air transfer resistance of a gas (R_{tw}), established through the sums of the transfer resistance in water (R_w), the transfer resistance in air (R_a), and the gas solubility (\alpha), is expressed in the equation:

R_{tw} = R_w + \alpha R_a



Including the relationship of the transfer resistance in air (R_a) with the gas transfer rate in air (k_a) in:

R_a = \displaystyle\frac{1}{ k_a }



The interaction of the transfer resistance in water (R_w) with the gas transfer rate in water (k_w) is described in:

R_w = \displaystyle\frac{1}{ k_w }



And the connection between the water to air transfer resistance of a gas (R_{tw}) and the total transfer rate of gas in water (k_{tw}) is detailed in:

R_{tw} = \displaystyle\frac{1}{ k_{tw} }



This provides the basis for establishing the relationship for the total transfer rate of gas in water (k_{tw}):

\displaystyle\frac{1}{ k_{tw} } = \displaystyle\frac{1}{ k_w } + \displaystyle\frac{ \alpha }{ k_a }

ID:(12243, 0)



Ocean-atmosphere transfer rate

Equation

>Top, >Model


The relationship of the transfer resistance between the ocean and the atmosphere can be expressed in terms of the transfer speeds in both mediums, corresponding to the inverse of the total transfer speed.

Thus, using the total transfer rate of gas in air (k_{ta}), the gas transfer rate in water (k_w), the gas transfer rate in air (k_a), and the gas solubility (\alpha), the following relationship is established:

\displaystyle\frac{1}{ k_{ta} } = \displaystyle\frac{1}{ k_a } + \displaystyle\frac{1}{ \alpha k_w }

\alpha
Gas solubility
-
9406
k_a
Gas transfer rate in air
m/s
9407
k_w
Gas transfer rate in water
m/s
9405
k_{ta}
Total transfer rate of gas in air
m/s
9447
k_w = ( u_a - u_w )* beta * Sc ^ n k_a = D / delta_c C_w = alpha * C_a R_a = 1/ k_a R_w = 1/ k_w R_tw = 1/ k_tw R_ta = 1/ k_ta R_ta = R_a + R_w / alpha R_tw = R_w + alpha * R_a 1/ k_ta = 1/ k_a + 1/( k_w * alpha ) 1/ k_tw = 1/ k_w + alpha / k_a R_taDnbetaC_aC_w0alphak_ak_wdelta_cSck_tak_twR_aR_wu_wu_aR_tw

The relationship involving the air to water transfer resistance of a gas (R_{ta}), determined by the combination of the transfer resistance in water (R_w), the transfer resistance in air (R_a), and the gas solubility (\alpha), is formulated in the equation:

R_{ta} = R_a + \displaystyle\frac{1}{ \alpha } R_w



This includes the relationship of the transfer resistance in air (R_a) with the gas transfer rate in air (k_a) expressed in:

R_a = \displaystyle\frac{1}{ k_a }



Additionally, the interaction of the transfer resistance in water (R_w) with the gas transfer rate in water (k_w) is explained in:

R_w = \displaystyle\frac{1}{ k_w }



And the connection between the air to water transfer resistance of a gas (R_{ta}) and the total transfer rate of gas in air (k_{ta}) is specified in:

R_{ta} = \displaystyle\frac{1}{ k_{ta} }



These elements together provide the basis for defining the relationship for the total transfer rate of gas in air (k_{ta}):

\displaystyle\frac{1}{ k_{ta} } = \displaystyle\frac{1}{ k_a } + \displaystyle\frac{1}{ \alpha k_w }

ID:(12242, 0)