L3 - Basic Dynamics

3.1 Horizontal Force Balance - Geostrophic Balance

• Away from boundaries and on timescales longer than a day, the atmosphere and ocean exhibit geostrophic balance, which is a state where the Coriolis force balances the pressure gradient force in large-scale flows.

• Geostrophic Balance Equation:

  • Pressure gradient force + Coriolis force = 0

• Geostrophic Wind Definition:

  • In geostrophic balance, fluids flow along contours of constant pressure, rather than directly from regions of high pressure to low pressure, due to the effects of the Coriolis force.

• Wind deflection:

  • Northern Hemisphere: Flow is deflected to the right due to the Coriolis effect, resulting in clockwise flow around high-pressure systems and counterclockwise flow around low-pressure systems.

  • Southern Hemisphere: Flow is deflected to the left, causing counterclockwise flow around high-pressure systems and clockwise flow around low-pressure systems.

• Represents the balance between forces acting on fluid parcels, significantly influencing weather patterns and ocean currents.

• Implications: Wind spirals around high and low-pressure centers in the atmosphere, leading to the formation of cyclones and anticyclones, which are critical in understanding weather systems.

3.2 Vertical Force Balance - Hydrostatic Balance

• The primary balance in the vertical direction is between the pressure gradient force and gravity, allowing for the stability of the atmosphere and ocean depths.

• Hydrostatic Balance Equation:

  • Pressure gradient force = Gravity

• Pressure equation:

  • Pressure at height z = weight of the air column above a unit area, reflecting how atmospheric pressure changes with altitude.

  • The decrease in pressure with height leads to variations in temperature and humidity at different altitudes.

• Key Pressure Value:

  • Sea level pressure: 1.013 x 10^5 Pa (1013 hPa = 1 atm), which serves as a benchmark for many meteorological measurements.

• Pressure decreases exponentially with height, primarily due to the diminishing weight of the air column above, which defines atmospheric structure and influences weather systems and climate.

3.3 Equation of State for a Dry Atmosphere

• All gases adhere to the Ideal Gas Law:

  • ( R = \frac{Rg}{ma} = 287 J kg^{-1} K^{-1} )

• Density of air under standard conditions is dependent on temperature and pressure results in a significant influence on flight dynamics, weather patterns, and climate behavior.

• Density Profile:

  • Exponential decrease in pressure with height is described by the e-folding scale H. For the troposphere: T0 ~ 250K and H = 7.3 km, illustrating variability in density with altitude that is fundamental to understanding atmospheric thermodynamics.

3.4 Equation of State for the Ocean

• Density of seawater varies depending on temperature (T), salinity (S), and pressure (P), which are critical factors in oceanic behavior and circulation patterns.

• Mean density of seawater: 1035 kg m^{-3} with variations of less than 7%, highlighting its relative consistency in oceanic studies.

• Pressure in Ocean:

  • At 10 m depth, the pressure exerts about 105 Pa or approximately 1 atm.

  • Pressure increases approximately linearly with depth due to the weight of the overlying water, which influences the physical and biological processes within the ocean.

• Salinity:

  • A measure of dissolved salts, typically around 34.5 g/kg (34.5 psu), which plays a crucial role in determining the density of seawater and affects ocean stratification and circulation dynamics.

3.5 Potential Temperature

• Potential Temperature (θ): Represents the temperature a fluid parcel would have if brought adiabatically to the surface without any heat transfer losses during ascent. It is crucial for understanding buoyancy in convective processes.

• As fluid moves from the surface upwards, expansion occurs and it cools without heat transfer, making potential temperature a conserved quantity in the absence of heat exchange.

• Fluid parcels retain potential temperature unless heat is added, providing a key insight into atmospheric stability and convection.

3.6 Thermal Wind Balance

• Combines geostrophic and hydrostatic balance to understand how geostrophic flow varies with height, which is essential for weather prediction and understanding climatic phenomena.

• Thermal Wind Equations: Describe how horizontal flow changes with height when horizontal density gradients exist, related to variations in temperature and pressure affects on wind patterns. This relationship is critical for predicting storm trajectories and intensities.

• For oceans, this understanding is crucial in regions like the Gulf Stream, where temperature and density differences are pronounced, affecting global climate patterns.

• Implications for the Atmosphere:

  • Zonal wind must increase with height to maintain thermal balance, indicating that changes in temperature gradients directly affect wind speed and direction, influencing weather systems globally.

Summary of Key Points

• Large-scale ocean and atmospheric circulations can be effectively modeled with geostrophic and hydrostatic balance equations, essential for meteorology and oceanography.

• Atmospheric pressure and density decay exponentially with height; ocean pressure behaves more linearly with depth, which is critical when analyzing marine environments.

• Density in the ocean is influenced by salinity and temperature variations, which drive thermohaline circulation patterns that regulate Earth's climate.

• In fluid dynamics without buoyancy, potential temperature and salinity remain conserved, facilitating prediction of fluid movement in various geophysical phenomena.

• The thermal wind equations clearly illustrate how flow is impacted by horizontal temperature and density gradients, particularly regarding height variations, underscoring the interconnectedness of atmospheric and oceanic dynamics.