 # (Introduction to) Earthquake Energy Balance

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(Introduction to) Earthquake Energy Balance
Mechanical energy, surface energy and the Griffith criteria Seismic energy and seismic efficiency The heat flow paradox Apparent stress drop The simplistic view

Earthquake energy balance: related questions
Are faults weaker or stronger than the surrounding crust? Do earthquakes release most, or just a small fraction of the strain energy that is stored in the crust?

Earthquake energy balance: Griffith criteria
The static frictionless case: UM is the mechanical energy. UA is the potential energy of the external load applied on the system boundary. UE is the internal elastic strain energy stored in the medium US is the surface energy. crack extends if: crack at equilibrium if crack heals if:

Earthquake energy balance: dynamic shear crack
Here, in addition to UA, UE and US: UK is the kinetic energy. UF is the work done against friction. During an earthquake, the partition of energy (after less before) is as follows: where ES is the radiated seismic energy.

Earthquake energy balance: dynamic shear crack
Since earthquake duration is so small compared to the inter seismic interval, the motion of the plate boundaries far from the fault is negligible, and UA=0. Thus, the expression for the radiated energy simplifies to: Question: what are the signs of UE, US and UF? Let us now write expressions for UE , US and UF .

Earthquake energy balance: elastic strain energy
To get a physical sense of what UE is, it is useful to consider the spring-slider analog. The reduction in the elastic strain energy stored in the spring during a slip episode is just the area under the force versus slip curve. For the spring-slider system, UE is equal to:

Earthquake energy balance: elastic strain energy
Similarly, for a crack embedded within an elastic medium, UE is equal to: where 1 and 2 are initial and final stresses, respectively, and the minus sign indicates a decrease in elastic strain energy.

Earthquake energy balance: frictional dissipation and surface energy
The frictional dissipation: where F is the friction, V is sliding speed, and t is time. Frictional work is converted mainly to heat. The surface energy: where  is the energy per unit area required to break the atomic bonds, and A is the rapture dimensions. Experimental studies show that  is very small, and thus surface energy is very small compared to the radiated energy (but not everyone agrees with this argument).

Earthquake energy balance: the simplest model
Consider the simplest model, in which the friction drops instantaneously from 1 to 2. In such case: F=2, and we get:

Earthquake energy balance: seismic efficiency
We define seismic efficiency, , as the ratio between the seismic energy and the negative of the elastic strain energy change, often referred to as the faulting energy. which leads to: with  being the static stress drop. While the stress drop may be determined from seismic data, absolute stresses may not.

Earthquake energy balance: seismic efficiency
The static stress drop is equal to: where G is the shear modulus, C is a geometrical constant, and (the tilde) L is the rupture characteristic length. The characteristic rupture length scale is different for small and large earthquakes. For small earthquakes, and Combining this with the expression for seismic moment we get: Both M and r may be inferred from seismic data.

Earthquake energy balance: seismic efficiency
Stress drops vary between 0.1 and 10 MPa over a range of seismic moments between 1018 and 1027 dyn cm. Figure from: Schlische et al., 1996

Earthquake energy balance: seismic efficiency
constraints on absolute stresses: In a hydrostatic state of stress, the friction stress increases with depth according to: where  is the coefficient of friction, g is the acceleration of gravity, and c and w are the densities of crustal rocks and water, respectively. Laboratory experiments show: Byerlee, 1978

Earthquake energy balance: seismic efficiency
Using: , the coefficient of friction = 0.6 c, rock density = 2600 Kg m-3 w, water density = 1000 Kg m-3 g, the acceleration of gravity = 9.8 m s-2 D, the depth of the seismogenic zone, say 12x103 m We get an average friction of: and the inferred seismic efficiency is:

Earthquake energy balance: seismic efficiency
So, the radiated energy makes only a small fraction of the energy that is available for faulting. Based on this conclusion a strong heat-flow anomaly is expected at the surface right above seismic faults.

Earthquake energy balance: the heat flow paradox
At least in the case of the San-Andreas fault in California, the expected heat anomaly is not observed. A section perpendicular to the SAF plane: Figure from: Scholz, 1990 The disagreement between the expected and observed heat-flow profiles is often referred to as the HEAT FLOW PARADOX.

Earthquake energy balance: the heat flow paradox
A section parallel to the SAF plane: Figure from: Scholz, 1990

Earthquake energy balance
The assumptions underlying the ''simple model'' are: Instantaneous drop from static to kinetic friction, and constant friction during slip. Uniform distribution of slip and stresses. Zero overshoot. Constant sliding velocity. No off fault deformation. The first point means that continuity is violated...

Earthquake energy balance
Other conceptual models: constant friction slip weakening quasi-static The simple model. The slip-weakening model. Significant amount of energy is dissipated in the process of fracturing the contact surface. In the literature this energy is interchangeably referred to as the break-up energy, fracture energy or surface energy. A silent (or slow) earthquake - no energy is radiated.

Earthquake energy balance
In reality, things are probably more complex than that. We now know that the distribution of slip and stresses is highly heterogeneous, and that the source time function is quite complex.

Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
Radiated energy and seismic moment of a large number of earthquakes have been independently estimated. It is interesting to examine the radiated energy and seismic moment ratio. Figure from: Kanamori, Annu. Rev. Earth Planet. Sci., 1994

Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
Remarkably, the ratio of radiated energy to seismic moment is fairly constant over a wide range of earthquake magnitudes. Figure from: Figure from Kanamori and Brodsky, Rep. Prog. Phys., 2004

Earthquake energy balance: radiated energy versus seismic moment and the apparent stress drop
What is the physical interpretation of the ratio ES to M0? Recall that the seismic moment is: and the radiated energy for constant friction (i.e., F = 2): Thus, ES/M0 multiplied by the shear modulus, G, is simply: This is often referred to as the 'apparent stress drop'.