Governing Thermodynamic Description of Sea Surface Temperature

To understand Sea Surface Temperature (SST) mathematically, we must view it as the thermodynamic result of the energy balance at the interface between the ocean and the atmosphere. In physical oceanography, SST is not a single value but a vertical profile. We differentiate between the Skin Temperature \(\text{SST}_{\text{skin}}\) and the Bulk Temperature \(\text{SST}_{\text{bulk}}\).

Temperature as a Prognostic Variable

The ocean temperature field is a continuous function of space and time: $$T = T(x, y, z, t)$$ where,

• \(x, y\) are horizontal coordinates,
• \(z\) is vertical depth (positive downward),
• \(t\) is time.

Temperature evolution in the ocean is governed by energy conservation.

Heat Conservation Equation (First Law of Thermodynamics)

The local rate of change of temperature is governed by the heat equation: $$ \rho c_p \frac{\partial T}{\partial t} = - \nabla \cdot \mathbf{F}_{\text{heat}} + Q $$ where:

• \(\rho\) is seawater density,
• \(c_p\) is specific heat capacity,
• \(\mathbf{F}_{\text{heat}}\) is the heat flux vector,
• \(Q\) represents internal heat sources (usually negligible near the surface).

Decomposition of Heat Flux

The heat flux consists of advective and diffusive components: $$ \mathbf{F}_{\text{heat}} = \rho c_p \mathbf{u} T - k \nabla T $$ where:

• \(\mathbf{u}\) is the fluid velocity,
• \(k\) is thermal conductivity (or turbulent diffusivity).

Substituting: $$ \frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T = \nabla \cdot (\kappa \nabla T) $$ This is the advection–diffusion equation for temperature.

Near-Surface Vertical Structure

Near the ocean surface, vertical gradients dominate over horizontal gradients. We simplify: $$ \frac{\partial T}{\partial t} \approx \frac{\partial}{\partial z} \left( \kappa \frac{\partial T}{\partial z} \right) $$ This approximation is valid within the upper few centimeters, where molecular and turbulent diffusion dominate.

Boundary Condition at the Air–Sea Interface

At the surface \((z = 0)\), the vertical heat flux is set by the net air–sea heat exchange: $$ - k \frac{\partial T}{\partial z}\Big|_{z=0} = Q_{\text{net}} $$ where: $$Q_{\text{net}}= Q_{\text{sw}} - Q_{\text{lw}} - Q_{\text{sens}} - Q_{\text{lat}}$$ This boundary condition couples ocean thermodynamics to the atmosphere.

Formation of the Skin Layer

In the uppermost micrometers, turbulent mixing vanishes and molecular diffusion dominates. Under quasi-steady conditions: $$ \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right)= 0 $$ which integrates to a linear gradient: $$\frac{\partial T}{\partial z} = -\frac{Q_{\text{net}}}{k}$$

Skin–Bulk Temperature Difference

Integrating over the skin layer thickness (\delta): $$T_{\text{skin}} - T_{\text{bulk}} = - \int_0^{\delta} \frac{Q_{\text{net}}}{k}, dz$$ which yields: $$ \boxed{ \Delta T = T_{\text{skin}} - T_{\text{bulk}} \approx - \frac{Q_{\text{net}}}{\rho c_p u_*} } $$ where \(u_*\) is the friction velocity, representing turbulent exchange.

This equation explains why:

• low wind → strong skin cooling,
• high wind → reduced skin–bulk difference.

Definition of Satellite-Relevant SST

Thermal infrared satellites sense radiation emitted from the skin layer, therefore: $$T_{\text{sat}} \equiv T_{\text{skin}} = T(z \rightarrow 0^+)$$ This is a boundary value, not a vertically averaged temperature.

Radiative Expression of Skin Temperature

The skin temperature enters the radiation field via Planck’s law: $$ B(\lambda, T_{\text{skin}}) = \frac{2hc^2}{\lambda^5} \frac{1}{\exp\left(\frac{hc}{\lambda k T_{\text{skin}}}\right)-1} $$ The emitted ocean radiance is: $$L_{\text{surf}}(\lambda) = \varepsilon(\lambda) B(\lambda, T_{\text{skin}})$$

Atmosphere–Ocean Coupled Observation Equation

Including atmospheric effects: $$L_{\text{TOA}}(\lambda) = \tau(\lambda) \varepsilon(\lambda) B(\lambda, T_{\text{skin}}) + L_{\text{atm}}^{\uparrow}(\lambda)$$ This is the forward physical model linking thermodynamics to satellite observations.

Observed Quantity and Inversion Target

The satellite measures channel-integrated radiance: $$L_c = \int L_{\text{TOA}}(\lambda) R_c(\lambda) d\lambda$$ The retrieval problem is: $$ \boxed{ \text{Given } L_c \Rightarrow \text{Estimate } T_{\text{skin}} } $$ This is an inverse problem, constrained by thermodynamics, radiative transfer, and statistics.

Conceptual Closure

Starting from the heat equation, we have shown that:

• SST is governed by energy conservation
• The skin temperature emerges from surface boundary conditions.
• Satellites sense a thermodynamic boundary layer.
• SST retrieval is fundamentally a physics-based inversion.

This framework ensures that SST is understood not as a “product”, but as the solution of a coupled thermodynamic–radiative system.