Hydrostatic Balance, Temperature and Thickness

• Hydrostatic Relationship in terms of temperature
• Substitute from equation of state to remove density:
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• Integrating from sea level with pressure p₀ to z:
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• Can't integrate precisely without knowing temperature variation with height
• Temperature variations in troposphere not large relative to absolute temperatures
• Decrease is approximately linear
• Can assume constant temperature with approximate tropospheric average without much error
• Relationship becomes:
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• Define scale height: then:
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• Scale height is the height at which pressure decreases by about 1/e
• Value of about 8 km for tropospheric average temperatures
• Under assumption of constant temperature, density also decreases at comparable rate:
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• Pressure as Vertical Coordinate
• Pressure is monotonic function of height, so can be used as vertical coordinate
• For tropospheric processes g and R can be considered constant
• Integrate hydrostatic balance equation upward
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• Define pressure-weighted average layer temperature:
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• Then layer thickness is proportional to average layer temperature (hypsometric equation):
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• Geopotential and Geopotential Height
• Geopotential defined as work required to lift 1 kg mass from sea level to height z:
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• Since g is nearly constant in troposphere
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• Geopotential height is defined as:
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• Since in troposphere gravitational acceleration is almost constant at sea level value, g₀